Coordinate Systems

CS 1

Concepts of primary interest:

The line element

Coordinate directions

Area and volume elements

Sample calculations:

Coordinate direction derivatives

Velocity and acceleration in polar coordinates

Application examples:

Velocity and acceleration in spherical coordinates

**** add solid angle

Tools of the Trade

Changing a vector

Area Elements: dA = 1 2dr dr

*** TO Add *****

Appendix I – The Gradient and Line Integrals

Coordinate systems are used to describe positions of particles or points at which quantities are to be defined

or measured. They are often used as references for specifying directions. The coordinate system or reference

frame is used extensively in describing the physical problem or situation, but it is not a part of the problem. No

physical result can depend on the choice of coordinates. The coordinate system is a passive aid to the

observer, and it may be chosen or adjusted to suit the purposes of the observer. Problem statements may use a

coordinate system as a convenience, but no physical problem comes with axes glued to it. We add them to

facilitate the description of the problem. Once the coordinates have been chosen for a problem and the

description has been started, further changes are usually not advised as a complicated transformation scheme is

often required to translate information stated relative to one set of coordinates into a form suitable for use in

another set of coordinates.

A system of coordinates for three dimensions assigns an ordered triplet of numbers [(x, y, z) or (q1,q2,q3)]

to each point in space. Three such coordinate systems are commonly used by undergraduate physics majors:

Cartesian, cylindrical and spherical. A common characteristic of these systems is that they are locally

orthonormal coordinate systems. This phrase means that each coordinate system specifies three mutually

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perpendicular (orthogonal and unity normalized) directions at every point in space. An infinitesimal

displacement along one coordinate direction is independent of small displacements along the other coordinate

directions because their directions are mutually perpendicular. For example, in Cartesian coordinates, a

displacement in the x direction does not change the y or z coordinate.

Each system is to be discussed in a parallel fashion to emphasize their common features and their

distinguishing characteristics.

Rene Descartes (1596-1650): French scientific philosopher who

developed a theory known as the mechanical philosophy. This

philosophy was highly influential until superseded by Newton’s

methodology. Descartes was the first to make a graph, allowing a

geometric interpretation of a mathematical function and giving his

name to Cartesian coordinates. Eric W. Weisstein @

scienceworld.wolfram.com/biography/Descartes.html

Cartesian Coordinates

To understand a coordinate system, you must know its relation to the Cartesian coordinate system, the

representation of the position vector, the shapes of the constant coordinate surfaces, the three independent

coordinate directions, and the line element represented as d or dr

. For this reason, the Cartesian system is

studied first. The relations between the coordinates of a Cartesian system and those of a second Cartesian

system with the same origin and axes directions are: x' = x, y' = y, and z' = z.

(A more interesting set of transformations is used to relate one set of Cartesian coordinates to another

Cartesian set with a different origin or orientation. That problem is studied in a second semester course in

mechanics.)

Constant Coordinate Surfaces: The constant coordinate surfaces are planes parallel to the plane defined

by the other two axes. For example, x = a is a plane parallel to the y-z plane that is perpendicular to the x axis at

the point (a, 0, 0). The point (a, b, c) is located at the intersection of the planes x = a, y = b, and z =c. You

should sketch some constant coordinate planes illustrating the intersections of pairs and triplets of such planes.

Coordinate Orbits: We define a coordinate orbit as the locus of points mapped as one coordinate runs

through its full range in the positive sense while the other coordinates are held fixed. An x-orbit is an infinite

line parallel to the x-axis that passes through the x = 0 plane at (0, y, z).

Position Vector: The position vector for a point P is the displacement from the origin to that point. The

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Cartesian position vector corresponding to the point P = (xP, yP, zP) is ˆˆ ˆP P P Pr x i y j z k

.

Coordinate Directions: One can find the coordinate directions by examining the change in position due to

a small positive variation in one coordinate while the other coordinates are kept fixed. Imagine the Cartesian

coordinate axes and a point (x, y, z) hanging in otherwise empty space. Increase each coordinate in turn by a

small positive increment to visualize each of the independent coordinate directions { ˆ ˆ ˆ, ,x y z }(also known as:{

ˆˆ ˆ, ,i j k }; { ˆ ˆ ˆ, ,x y ze e e } or { 1 2 3ˆ ˆ ˆ, ,e e e }).

The x direction is the direction a point is displaced if its x coordinate is given a small positive

increment while its y and z coordinates are held fixed.

0

ˆ ˆˆ ˆ ˆ ˆ( , , ) ( , , ) ˆˆ( , , ) ( , , )dx

x dx i y j z k x i y j z kr x dx y z r x y zx i

r x dx y z r x y z dxLimit

FORGET the equation! It is the picture that you need. Imagine the axes and point hanging out in space.

In your mind, move the point from (x, y, z) to (x + x, y, z). In what direction did the point move?

Line Element: The next vital quantity is the line element which is found as the displacement from the

point (x, y, z) to the point (x + dx, y + dy, z + dz) at which each of the coordinates has been given an

infinitesimal increment.

ˆˆ ˆ( , , ) ( , , )d dr r x dx y dy z dz r x y z dx i dy j dz k

Area and volume elements are built up from the mutually orthogonal components of the line element. For

an area element with its normal in the x direction, x is fixed, and dAx = dy dz. The area element is just the

product of the two perpendicular components of the line element. All three components of the differential of

area are summarized as:

ˆ ˆˆ ˆ ˆ ˆx y zdA dA i dA j dA k dy dz i dz dx j dx dy k

(There are other notations for dA such as dS, and 2d r .)

Note that the direction of an area element is defined to be one of its normal directions. For a closed surface, the

convention is to choose the outward directed normal. For the area element dA

, the convention is that one takes

the cross product of the each pair of the dir4ected components of the line element in right hand rule order.

ˆ ˆˆ ˆ ˆ ˆ ˆˆ ˆdxi dy j dy j dz k dz k dx idA dy dz i dz dx j dxdy k

Finally, we get the volume element by computing the product of the three orthogonal components of the line

element. A volume element can be swept out by taking a small area element and moving it a small distance in

the direction of its normal, or it can be computed as the triple vector product of the line elements component

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vectors.

ˆ ˆˆ ˆ ˆ ˆx y zdV dxi dA i dy j dA j dz k dA k dx dy dz

ˆˆ ˆ( )dV dxi dy j dz k dV = dx dy dz

Everything is constructed from the components of the line element. Note the coordinate cube (volume element)

has a small coordinate corner at (x,y,z) and a large coordinate corner at (x + dx, y + dy, z + dz). The components

of the line element ˆˆ ˆ{ , , }dxi dy j dz k are drawn from the small coordinate corner and highlighted. Then, the

remaining 9 edges are added. Volumes and areas are easy because the components of the line element are

mutually perpendicular.

Exercise: Consider an area element ˆydA j . Compute the volume that is swept out by the area as it is given each

of the following displacements: , and ˆˆ ˆdx i dy j dz k . Prepare sketches.

Direction Cosines: A general direction is expressed as:

ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ( ) ( ) ( ) (cos ) (cos ) (cos )e i e i j e j k e k i j k

where cos, cos and cos are the direction cosines of with respect to the three coordinate directions. That is

cos = ˆ ˆi e the cosine of the angle between the direction of e and that of i , the x direction.

y

x

z

i j

k

dV = dx dy dz

dAx = dy dz

dAz = dx dy

ˆdxi ˆdy j

ˆdz k

ˆdz k

ˆdy j

ˆdy j

ˆdxi

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Cylindrical Coordinates

Orientation relative to the Cartesian standard system: The origins and z axes of the cylindrical system and of

the Cartesian reference are coincident. The cylindrical radial coordinate is the perpendicular distance from the

point to the z axis. The angle is the angle between the x axis and the projection of the position vector in the x-y

plane. Coordinate ranges: 0 r < ∞, 0 < 2π, and -∞ < z < ∞.

NOTATION ALERT: The radial coordinate r represents the distance from the axis in the cylindrical system.

It is chosen to coincide with the standard notation used for 2D polar coordinates. This notation has a potential

pitfall as 2 2r r zr , and it can be confused with the spherical radial coordinate r = r

, the distance

from the origin. Stay Alert! Why do we use this ambiguous notation? Unfortunately, there is no uniformly

adopted notation that avoids the potential for confusion. Some authors use or perhaps s for the cylindrical

radial coordinate, but just as many use r. The symbols and s are also multiply assigned as charge or mass

densities or as distance. It is therefore the choice of this author to follow the common practice of using r. Keep

your head in the game. As of this moment, the Mech II and E&M II texts used different notations for the

cylindrical radial coordinate. Every common symbol is used to represent two or more concepts. Learn to map

symbols to their concepts at all times. Do not read symbols as their symbol name; read them as the concepts that

they represent.

Relation of Cylindrical Coordinates to Cartesian coordinates:

2 2x yr = 1tan yx

z = z

x = r cos y = r sin z = z

Constant coordinate surfaces:

r = constant: an infinite circular cylinder concentric with the z axis.

= constant: a half infinite plane starting on and including the z axis and the ray

= constant in the z = 0 plane.

z = constant: an infinite plane perpendicular to the z axis at the point z = constant on

that axis.

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2 2

1tan yx

r x y

z z

cos

sin

x r

y

ˆˆr r r z k

ˆ ˆˆ cos sin

ˆ ˆ ˆsin cos

ˆ ˆ

r i j

i j

k k

Exercise: Show that ˆˆ andr are orthogonal. Geometrically speaking, the direction of a radial line running to the

edge of a circle is orthogonal to a tangent line to that circle at the same point. The directions are locally

orthogonal.

dz

(r+dr) d

r d dr

A cylindrical coordinate “cube”

Note that the radial faces have different areas.

y

ˆzk

r

k

x

z

ˆr r

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Another cylindrical coordinate cube.

line element: ˆˆˆdr dr r r d dz k

The three components of the line element are to be drawn

from the small coordinate corner at (r,,z) and appear

highlighted. The remaining sides are dashed. The volume

and area elements are:

dV = dr rd dz

dAr = rd dz dA = dr dz dAz = dr rd

Cylindrical Coordinate Orbits:

The coordinate orbits are the loci of points that are found by varying one coordinate over its full range while the

other two coordinates are held fixed.

r orbit: a line perpendicular to the z axis at z and that has a projection into the z = 0 plane that makes an

and with respect to the x direction. (0 r < ∞)

orbit: a circle of radius r with center at z = z on the z axis. (0 < 2π)

z orbit: a line parallel to the z axis through (r, in the x-y plane. z

Coordinate Directions and Line Element Components:

The cylindrical radial component of the line element is found by considering the displacement from ( , , )r r z

to ( , , )r r dr z for dr > 0. This displacement has magnitude dr and is directed perpendicular to and away

from the z axis at the point (r, , z).

The cylindrical phi direction is found by considering the displacement from ( , , )r r z to ( , , )r r d z

for

d> 0. This displacement has magnitude r d and is directed in the positive sense along a tangent to the circle

of radius r that is centered on the z axis and that passes through the point (r, , z). The z component of the line

element ˆdz k is identical to that for the Cartesian case.

Important Consideration: Only locally orthonormal coordinate systems are to be studied. In each of these

systems, the three coordinate orbits through a point cross at right angles to one another. The three coordinate

directions are tangent to the coordinate orbits and hence are mutually orthogonal at each point P in space.

y

ˆzk

dr r

dz k

x

z

ˆr r

rd

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They form a mutually orthogonal triad, the same configuration as ˆˆ ˆ, ,i j k except that in cylindrical and

spherical coordinates, the orientation of the triad varies point to point. Each coordinate direction is to be

represented as perfectly straight line segment of length one. The curvature of the coordinate orbits is an

important feature. Practice sketching line elements and coordinates cubes to show this curvature for cylindrical

and spherical coordinates while using laser-straight line segments directed tangent to the local arc of the

coordinate orbit to represent the local coordinate directions. In the limit that the differentials become small,

line elements approach straight lines, area elements are almost rectangular, and volume elements

approximately rectangular parallelepipeds.

The cylindrical position vector for the point P = (r, z) is ˆˆP Pr r r z k where Pr is defined at P. Note that

r is position dependent while k is a globally constant vector. The position vector is the displacement from the

origin to the point P (The P subscript is to be suppressed.). Why is no part needed to represent the position

vector? Note that Pr means r defined at the point P.

In cylindrical and spherical coordinates, the coordinate directions are functions of the angular coordinates

only. The direction k is a global constant in cylindrical coordinates; all three directions are coordinate

dependent in spherical coordinates.

Line Elements and Angular Coordinates For the systems that we study, the coordinate point moves along an arc of a circle when an angular

coordinate is varied while the other coordinates are held fixed. The distance traveled along the arc is the

increment of the angle times the radius of the arc. (Review the definition of radian measure for angles.)

For the case above, the increment of the angle was d and the radius of the path (or orbit) of the coordinate

point was r.

Radian measure is defined in terms of an angle with is vertex at the center of a circle. The angle is the ratio

of the arc length that subtends the angle to the radius of the circle sR where s is the length of the arc

segment between A and B. The distance around the circle is 2R so the total angle around the circle is 2

radians.

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It follows that a particle has a tangential speed v = /t R = R and the tangential acceleration is at = tR

= R.

The line element in cylindrical coordinates is:

ˆˆˆ( , , ) ( , , )d dr r r dr d z dz r r z dr r r d dz k

Cylindrical Area Elements:

We will discuss the area element in 2-D polar coordinates as a first step.

dA = dr rd ˆˆ ˆ ˆ ˆˆ cos sin sin cosr i j i j

For small dr and d, the area element is approximately a rectangle with area: dA = dr (r d. Picture the shape

of the area element as dr and d become small. In that limit, two of the sides approach ˆdr r and ˆr d , and as

the r and directions are orthogonal (perpendicular), the area is just the product of the lengths of the sides.

For the cylindrical coordinate system, there are three basic area elements, each with its normal along one of the

A

B

x

y Circle of Radius R

O

s

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CS 10

three coordinate directions. We have found the element dAz = dr r d which has the z direction as its normal.

The component with radial normal dAr = r ddz has sides consisting of the line elements for the other two

directions. The final element dA has ˆdr r and ˆdz k as sides; hence dA = dr dz

The Volume Element:

The volume element is the product of the three components of the line element. dVcyl = dr rd dz. Recall that

the three components of the line element are mutually perpendicular. They are three edges of a coordinate

"cube" or volume element.

Order of Coordinates: A General Rule A conventional order has been assigned to the coordinates used in

each system. They are to be listed in a right-hand-rule ordered sequence. That is r, , z for cylindrical

coordinates. It follows that the cross product of the first two coordinate directions taken in the conventional

order yields the third direction.

ˆ ˆ ˆˆ ˆ ˆˆ ˆ ˆ, andr k k r k r

The three coordinate directions at any point are orthogonal and normalized. The radial direction at one point

may not be orthogonal (perpendicular) to the phi direction at another point, but it will be orthogonal to the phi

direction at that same point. The coordinate directions are therefore called locally orthonormal. The Cartesian

system is the only one that has a set of coordinate directions that are globally orthonormal.

Spherical Coordinates

Relative Orientation as Compared to a Cartesian Reference:

The origins are coincident. The polar axis from which is measured and the Cartesian z axis are parallel. The

angle is measured between the Cartesian x axis and the projection of the position vector onto the x-y plane.

Relation of Spherical Coordinates to Cartesian coordinates:

2 2 2r x y z 1

2 2 2cos z

x y z

1tan y

x

x = r sin cos ; y = r sin sinz = r cos

Constant coordinate surfaces:

r = constant: A spherical surface of radius r centered on the origin.

= constant: A right circular cone with apex at the origin and half angle about the polar axis.

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= constant: A half infinite plane with the polar axis as one edge and including the line = constant in the

x-y plane.

x = r sin cos y = r cos sin z = r cos

The spherical coordinate directions can be mapped to the common directions at a

point on the earth’s surface. ˆ ˆˆ{ , , }r {up, south, east}.

ˆˆ ˆˆ sin cos sin sin cosr i j k ; ˆˆ ˆ ˆcos cos cos sin sini j k ; ˆ ˆ ˆsin cosi j

Spherical Coordinate Orbits:

r orbit: A fixed and ray from the origin to infinity. 0 r

orbit: A semicircle of longitude with radius r in the constant plane. 0

orbit: A circle of latitude of radius r sin at the intersection of the = constant cone with the spherical

surface of radius r concentric with the origin. 0 2

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Coordinate Directions and Line Element Components:

We will choose dr, d and d to be positive. The spherical radial component of the line element is found by

considering the displacement from ( , , )r r to ( , , )r r dr

. This displacement has magnitude dr and is

directed away along the , ray. The spherical theta component of the line element is found by considering the

displacement from ( , , )r r to ( , , )r r d

. This displacement has magnitude r d and is directed in the

positive sense (south) along a tangent to the circle of longitude that has radius r and lies in the constant plane.

The spherical component of the line element is found by considering the displacement from ( , , )r r to

( , , )r r d . This displacement has magnitude r sin d and is directed along the tangent to a circle of

latitude in the positive sense (east).

For a small increment in an angular variable, the tip of the position moves along a circular path, and it has a

magnitude equal to the radius of the 'path' times the change in angle. (Review the definition of radian measure

of angles.)

The Spherical Position Vector for the point P = (r,) is ( , , )r r = ˆr r where r is defined at P and

depends on and . While the radial coordinate r appears explicitly in ˆr r r , the dependence on the

coordinates and is hidden in r .

The line element in spherical coordinates is:

ˆ ˆˆ( , , ) ( , , ) sind dr r r dr d d r r dr r r d r d

Spherical Area Elements: As before, the three infinitesimal displacements: ˆdr r , ˆr d and ˆsinr d are

mutually perpendicular. The area elements are:

sinrdA r d r d sindA dr r d dA dr r d

2ˆ ˆ ˆ ˆˆ ˆsin sinrdA dA r dA dA r d d r r dr d r dr d

The Volume Element:

The product of the three components mutually perpendicular components of the line element is the volume

element. dVsph = r 2 sin d d dr.

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!!! A terrible drawing. Practice until you can do much better.

Exercise: Prepare a coordinate cube sketch for the spherical volume element. Draw each component of the

spherical line element originating from the small coordinate corner. Label and highlight those edges. Add the

remaining 9 edges.

Order of Coordinates: A General Rule

A conventional order has been assigned to the coordinates used in each system. They are listed in a right-

handed sequence. That is r,, for spherical coordinates. It follows that the cross product of the first two

coordinate directions taken in the conventional order yields the third direction.

Spherical Coordinate 'Cube'

r sin(d d

(r,)

r sin d

(r+dr,dd)

r d

dr

(r+dr) d

y

x

z

dr

r d

r sin d

(r+dr,+d,+d

(r,,

Spherical Coordinate Cube

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CS 14

ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ, andr r r

The three coordinate directions at any point are orthogonal and normalized. The radial direction at one point

may not be orthogonal (perpendicular) to the theta direction at another point, but it will be orthogonal to the

theta direction at the same point. The coordinate directions are therefore called locally orthonormal. The

Cartesian system is the only one that has a set of coordinate directions that are globally orthonormal.

Properties of (Locally Orthonormal) Coordinate Directions

In each coordinate system, the unit vectors form a right-handed locally orthonormal set. Note the order of

the directions matches the standard right-hand order of the coordinates for a point.

DIRECTION CARTESIAN CYLINDRICAL SPHERICAL

1e i r r

2e j

3e k k

Each set satisfies the relations:

1 1 2 2 3 3ˆ ˆ ˆ ˆ ˆ ˆ1 1 1e e e e e e

1 2 2 3 3 1ˆ ˆ ˆ ˆ ˆ ˆ0 0 0e e e e e e

1 2 3 2 3 1 3 1 2ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆe e e e e e e e e

1 1 2 2 3 3ˆ ˆ ˆ ˆ ˆ ˆ0 0 0e e e e e e

Integrations with Unit Vectors: The use of spherical or cylindrical coordinates with their associated unit

vectors often simplifies integrations required in physics. It is usually most efficient to evaluate all the inner (dot)

products and cross products using the native unit vectors and the relations above. However, if, after these

evaluations, the integration of a non-constant unit vector is required, you should replace that unit vector by its

representation in terms of the constant Cartesian unit vectors as they can be taken outside the integrals. [Ex.:

ˆ ˆ ˆsin cosi j .]

The integration of spatially dependent unit vectors is too horrible to be considered in polite society.

However, a few examples are to be included. The results are found in terms of differences involving coordinate

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directions evaluated at distinct locations that require some decoding in the case that the coordinate directions

that are position dependent.

SYSTEM POSITION r

LINE ELEMENT d dr VECTOR F

Cartesian ˆˆ ˆx i y j z k ˆˆ ˆdx i dy j dz k ˆˆ ˆx y zF i F j F k

cylindrical ˆˆr r z k ˆ ˆˆdr r r d dz k ˆˆˆr zF F F kr

spherical ˆr r ˆ ˆˆ sindr r r d r d ˆ ˆˆrF F Fr

Transforming the component representation of a vector: A vector has a distinct representation in each of the three

coordinates systems discussed.

ˆ ˆˆˆ ˆ ˆ ˆˆ ˆx y z r z rF F i F j F k F F F k F F Fr r

An individual component can be projected out of a vector representation using the inner product with the direction of

interest. For example, compute i F

.

ˆ ˆˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆx y z r z ri F i F i F j F k i F F F k i F F Fr r

ˆ ˆˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆˆ ˆx y z r z ri F F i i F i j F i k F i F i F i k F i F i F ir r

ˆˆˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆˆ ˆx r z ri F F F i F i F i k F i F i F ir r

The inner product of the x direction with a vector i F

= Fx, the x component of that vector. Adding the expressions

for the y and z components, the relations can be summarized by a set of matrix equations.

,,

ˆˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆˆ ˆ

ˆˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆˆ ˆ

x r r

y

z zCar cyl sphCart sphCart cyl

i r i i k i r i iF F F

F j r j j k F j r j j F

F F Fk r k k k k r k k

(Note that this equation demonstrates the inadequacy of using a triplet of numbers to represent a vector. The basis, the

set of coordinate directions, must be known in addition to the values of the components to uniquely specify the

vector. In the equation above, the triplets of components are supplemented with a subscript to identify the set of

coordinate directions. For physics applications, explicitly displaying the coordinate directions, as demonstrated in the

equation set above the matrix equations, is the best choice.

Exercise: Use the representations of the cylindrical directions in terms of the Cartesian directions to compute:

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,

ˆˆˆ ˆ ˆˆ

ˆˆˆ ˆ ˆˆ

ˆ ˆ ˆ ˆˆˆCart cyl

i r i i k

j r j j k

k r k k k

Exercise: Use the representations of the spherical directions in terms of the Cartesian directions to compute:

,

ˆˆˆ ˆ ˆˆ

ˆˆˆ ˆ ˆˆ

ˆ ˆ ˆ ˆˆˆCart sph

i r i i k

j r j j k

k r k k k

Exercise: Give the form of the transformation matrix that would be labeled with the subscript cyl,sph.

Derivatives of the Coordinate Directions

Note that the cylindrical radial coordinate is sometime labeled or s. Only r is to be used here. The azimuthal

angle is often labeled rather than . Be prepared to make the translations on the fly.

The Cartesian coordinate system is the only locally orthonormal system for which the coordinates directions are

constants. They are global constants, the same at every point in space.

For the cylindrical and spherical systems, the coordinate directions are not constant, but rather vary in direction as

one moves from position to position. It is worth noting, that the coordinate directions depend only on the angular

coordinates for the systems that we study (Cartesian, cylindrical, and spherical). This feature is not universal; it's just

something that works for our cases of interest. You should return to the defining process above if you ever use other

coordinate systems.

Cylindrical Coordinates: The coordinate directions in cylindrical coordinates and their relations to the

Cartesian coordinate directions are given below.

Notation Alert: The cylindrical coordinates are to be labeled r, and z. In other references, the

radial coordinate is often labeled as or s, and the azimuthal angle is often labeled .

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The direction is perpendicular to the radial direction (along the tangent line to the circular orbit at the point

and directed in the sense of increasing . The primed unit vectors are shown at a nearby point. Would the

coordinate directions change if r were increased to rdr with held fixed?

In order to find the derivatives of the unit vectors, we first notice that they depend on the angle and not on the

radial coordinate. Next translate all the unit vectors to the origin with their directions fixed.

ˆˆdr d

ˆ ˆd d r

ˆ ˆdr

d

ˆˆd

dr

ˆ ˆˆ cos sinr i j

ˆ ˆ ˆsin cosi j

ˆz k

ˆˆ

ˆ ˆ( )

dr d

d d r

The changes in the unit vectors for a change of angle d are easily read off the figure. The length of ˆdr is

just d because the tip of r is following a circular path of radius 1 as varies. The direction of each change is

x

ˆ( )r

ˆ( )r d ˆˆdr d ˆ( )d

ˆ( )

ˆ ˆd d r

d

d

y

x

y

ˆ( )r

ˆ( )

ˆ( )r d ˆ( )d

d

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along a tangent to the circle for which the original unit vector is a radius. We find: ˆ ˆˆ ˆand ( )dr d d d r .

Thus the coordinate directions depend only on in cylindrical coordinates and their derivatives are:

DIRECTION DERIVATIVES FOR CYLINDRICAL COORDINATES

ˆˆ ˆ ˆ ˆ; ;dr d

d dr k

constant

Spherical Coordinates:

ˆˆ ˆˆ sin cos sin sin cosr i j k

ˆˆ ˆ ˆcos cos cos sin sini j k

ˆ ˆ ˆsin cosi j

Make a clear figure for each direction and vary only one coordinate at a time. The three spherical

coordinate directions illustrated above are translated to originate at the origin with their directions held fixed to

arrive at the figure below. They depend on both and . The resulting change in the direction divided by the

change in the coordinate varied yields the partial derivative of that direction with respect to that coordinate in

the limit of an infinitesimal change.

Coordinate Systems

CS 19

: makes an angle with respect to the polar axis; its projected

length on the x-y plane is sin.

: lies at an angle below the x-y plane; its projected length on the

x-y plane is cos.

: lies in the x-y plane; its projected length on the x-y plane is 1.

ˆˆ ˆˆ sin cos sin sin cosr i j k

ˆˆ ˆ ˆcos cos cos sin sini j k

ˆ ˆ ˆsin cosi j

Vary the angle while holding fixed. The tips of the direction vectors and trace paths along a unit radius

circle of longitude that is a constant coordinate orbit. The direction does not change as is varied.

Exercise: Sketch ˆdr , ˆd and d given that is varied and is held fixed. What are the radii of the paths

traced by the tips of the directions? Give expressions for ˆdr , ˆd and d .

If is varied while is held fixed, and trace paths along circles of latitude with radii sin and cos. The

direction traces along the equatorial ( =

/2 ) circle of latitude. Note that the direction that lies in the x-y

plane that is perpendicular to can be expressed as a linear combination of and .

Exercise: Sketch ˆdr , ˆd and d given that is varied and is held fixed. What are the radii of the paths

traced by the tips of the directions? Give expressions for ˆdr , ˆd and d .

DIRECTION DERIVATIVES FOR SPHERICAL COORDINATES

ˆ r

ˆ

ˆ r ˆ

ˆ r ˆ

ˆ r ˆ

Coordinate Systems

CS 20

ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆ ˆsin

ˆˆ cos

ˆˆ0 sin cos

r r

r

r

Application: Acceleration in Polar Coordinates

***The label is used for in this section.

In polar coordinates, the position vector ˆr r r . The velocity of the particle is dr

v rdt

where the dot

notation is adopted to represent time derivatives because d by d(whatever) is a huge pain.

dot notation for time derivatives 2

2;

dr d rr r

dt dt , etc

The issue is that r is not constant as it represents the radial direction at the position of the particle. The

direction r depends on and hence on time by the chain rule.

ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆr

dr dr dr dr d drv r r r r r r r r r r r v r v

dt dt dt d dt d

This result identifies: rv r and v r r . The particle has a radial component of velocity if r, its

distance from the origin, is changing in time. As angular coordinate varies, the particle moves along the

direction tangent to the circle (arc of constant radius) through its position. If r is fixed, the only motion is

tangential, along the circular path. The only way to move and stay on a path is to move tangentially, in the

direction of the path. Motion perpendicular to the circular path (radial) would carry the particle off the path.

For the special case of circular motion r = 0 hence: ˆ ˆv v r

.

Continuing, the polar representation for acceleration is:

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆdv d

a r r r r r r r r r rdt dt

where, after another application of the chain rule, it results that ˆ r and thus:

Coordinate Systems

CS 21

2 2ˆ ˆˆ ˆ2 2 r

dva r r r r r r r r r v

dt

where and are called angular velocity and angular acceleration. That is:

2 2ˆ ˆ ˆˆ ˆ ˆ2 2r ra a r a r r r r r r r r r v

While the terms 2 ˆr r r and ˆ ˆorr r are often studied in a first physics course, the term

ˆ2 r is not thoroughly discussed. If you reflect for a moment, that term is active when an ice skater

changes her rate of spin by adjusting the positions of her arms. In the study of rigid body rotation, the r’s are

fixed so r vanishes.

Why is it so hard! Why not: andra r a r r ?

Consider a particle moving with constant velocity in polar coordinates. At time zero, it is at r = ro and = 0

with velocity vo in the y-direction. The acceleration must be zero as the velocity is constant.

Even though the particle has zero acceleration, r varies in time. 22o or r v t Approximating r by expanding

using the binomial theorem

(1/ 2)2 2 2

22 21 11 1

2 2o o o

o o o o oo o o

v t v t vr r v t r r r t

r r r

and comparing with 21

2o ox xx x v t a t suggests a radial acceleration of 2

o

o

v

r even for motion at constant

velocity. The adjustment 2

2 2 vr r r patches the nonsense introduced by the non-constant

coordinate directions and by coordinate orbits that are not straight lines. Consider the right half of the figure.

As the particle moves up with constant velocity, the theta component of velocity decreases because the theta

x

y

vt

ro

r

x

y

ro

r

r

v

v

Coordinate Systems

CS 22

direction changes. The term 2 r patches the damage done by the axes at the particle’s location changing

direction while the particle moves at constant velocity (with zero acceleration). See problem 32 for more

details.

All this funny business is just another reason for loving Cartesian coordinates. But then again, real

understanding springs from arguments of this type. If the apparent rel where is the rotation rate the

reference frame relative to an inertial coordinate system and rel is the particle's angular velocity relative to the

rotating forms, the 2 ˆr r and - ˆ2 r contributions due to the frame rotation rather than to the particle's

motion seem quite magical. These pieces are the centrifugal and Coriolis terms observed in a rotating system.

DO NOT think about this now. DO consider it carefully when these so called 'ghost' or inertial effects are

introduced in your mechanics or dynamics course.

A Meandering Mind Section: Dynamics in a rotating frame.

The acceleration by its nature is to be evaluated in an inertial frame. In a frame that rotates at relative to an

inertial frame, rel where is constant.

OBSERVES: 2 ˆˆ 2rel rel rel rela r r r r r

ACTUAL: 2 ˆˆ 2a r r r r r

SUBSTITUTE: rel Note that the rotating and inertial observes measure the same values for

, andr r r . IDENTIFY the extra pieces

2 ˆˆ 2rel rel rela r r r r r

2 2

2

2

ˆ ˆˆ ˆ ˆ2 2 2

ˆˆ ˆ2 2

ˆˆ ˆ2 2

rel rel rel rel

rel rel

rel rel

a r r r r r r r r r r

a a r r r r r

a a r r r r r

If the observer in the rotating frame attempts to use Newton's 2nd Law, he must impose special 'forces' for which

no entity can be found that exerts them.

Coordinate Systems

CS 23

2

2

:

:

ˆˆ ˆ2 2

ˆˆ ˆ2 2

rel rel

rel i relreal

forces i

rel i centrifugal Coriolisreal

forces i

m a m a m r r m r r m r

m a F m r r m r r m r

m a F F F

Recalling that the rotation axis is the cylindrical z axis, k

and the velocity of the particle relative to the

rotating frame is ˆˆrelv r r r , these inertial or ghost

forces can be represented as centrifugal rF m

and 2 ( )relCoriolisF m v

.

centrifugal: center fleeing centripetal: center seeking

Coriolis Effect: An inertial effect that mimics a force described by the 19th-century French

engineer-mathematician Gustave-Gaspard Coriolis in 1835.

Cartesian:

Position ˆˆ ˆr xi y j z k

Line Element ˆˆ ˆdr dxi dy j dz k

Velocity ˆˆ ˆv xi y j z k

Acceleration ˆˆ ˆa xi y j z k

Velocity squared 2 2 2 2v x y z

Cylindrical:

Position ˆˆr r r z k

Line Element ˆˆˆdr dr r r d dz k

Velocity ˆ ˆˆ ˆˆ ˆr zdr

dtv r r r z k v r v v k

Coordinate Systems

CS 24

Acceleration 2 ˆ ˆˆ ˆˆ ˆ2 r za r r r r r z k a r a a k

Velocity squared 22 2 2 2 2 2

r zv r r z v v v

Spherical:

Position ˆr r r

Line Element ˆ ˆˆ sindr dr r r d r d

Velocity ˆ ˆ ˆ ˆˆ ˆsin rdr

dtv r r r r v r v v

Acceleration

2 2 2

2

2

ˆsin

ˆ2 sin cos

ˆsin sin 2 cos

a r r r r

r r r

r r r

Velocity squared 2 22 2 2 2 2sin rv r r r v v v

Exercise: Divide the line element in each system by dt to develop the velocity in each system.

General (locally orthogonal) Curvilinear Coordinates: UNDER CONSTRUCTION

Coordinate systems for three-dimensional space label each point in the space with a triplet of values (q1,

q2, q3). A general 3-D coordinate system with coordinates q1, q2, q3 and locally orthogonal coordinate

directions 1 2 3ˆ ˆ ˆ, ,e e e has a line element which can be written in the form:

1 1 2 2 3 3 1 2 3( , , ) ( , , )dr d r q dq q dq q dq r q q q

1 1 2 2 3 31 2 3 1 1 2 3 2 1 2 3 3ˆ ˆ ˆ( , , ) ( , , ) ( , , )dr d h q q q dq e h q q q dq e h q q q dq e

Coordinate Systems

CS 25

The coordinate directions 1 2 3ˆ ˆ ˆ, ,e e e vary from point to point, but at any point are mutually (orthogonal)

perpendicular. The hallmark of these systems is that an infinitesimal change in one of the coordinates (say 2q to

2 2q dq ) does not change either of the other coordinates. Infinitesimal displacements in the coordinate

directions are independent (orthogonal/perpendicular) of one another. The coordinate direction 1e is defined

as:

1 11

1

1 1 2 3 1 2 3 1 2 3 1

1 2 3 11 1 2 3 1 2 3

ˆ( , , ) ( , , ) ( , , )ˆ

( , , )( , , ) ( , , )r q dq q q r q q q h q q q dq e

eh q q q dqr q dq q q r q q q

where dq1 is assumed to be positive. Compare this equation with the one for ˆˆ ( )x i on page 2. The direction

ie would result if qi were incremented. For example

2 22

2

1 2 2 3 1 2 3 1 2 3 2

1 2 3 21 2 2 3 1 2 3

ˆ( , , ) ( , , ) ( , , )ˆ

( , , )( , , ) ( , , )r q q dq q r q q q h q q q dq e

eh q q q dqr q q dq q r q q q

That is: the unit vector ie associated with qi is the direction in which the coordinate point moves when the

coordinate qi is given a small positive increment while the other coordinates are held fixed. The scale functions

hi(q1,q2,q3) squared are the elements of the metric for an orthogonal coordinate system and, in particular, the

length of dr

squared is:

2 2 221 1 2 3 1 2 1 2 3 2 3 1 2 3 3( , , ) ( , , ) ( , , )ds dr dr h q q q dq h q q q dq h q q q dq

Recalling that ds 2 = dx

2 + dy

2 + dz

2, the elements of the metric may be computed for locally orthonormal

systems from the definitions of the Cartesian coordinates (x,y,z) = (x1, x2, x3) in terms of the q's.

3 3 32 2 2

11 1 11 2 33 31 1 11 1 2 2 3 3

; ;k k k k k k

k k k

x x x x x xg h g h g h

q q q q q q

In terms of the metric, 3

2

, 1ij i j

i j

ds g dq dq

. The metric ijg is diagonal for locally orthogonal coordinate

systems. That means that 0ijg if i j . For example, the line element in spherical coordinates is:

ˆ ˆˆ sinspherical

dr dr r r d r d

2 2 22 sinds dr dr dr r d r d

and the non-zero metric elements are: 2 2 211 22 331, , (sin )g g r g r .

Coordinate Systems

CS 26

For the systems studied above:

System Coordinates h1 1e h2 2e h3 3e

Cartesian x, y, z 1 i 1 j 1 k

cylindrical r, , z 1 r r 1 k

spherical r, , 1 r r r sin

There are eleven locally orthonormal coordinate systems.

The local orthogonality facilitates the separation of partial differential equations into sets of ordinary differential

equations in these systems.

In more general systems, g ij = 3

1

k k

k i j

x x

q q

General Coordinate 'Cube' with some outwardly directed normals.

Coordinate Systems

CS 27

We consider a coordinate 'cube' with one corner at (q1,q2,q3) and the opposite corner at

(q1+dq1,q2+dq2,q3+dq3). In the limit that the dq's 0, the ie along each edge of the 'cube' converge to the

same three directions, the three coordinate directions at the limit point. [Note that - 2e drawn in the center of

the figure is 'out of the page'.]

The volume element is the product of the three sides:

1 2 31 2 3 1 2 3 1 2 3 1 2 3( , , ) ( , , ) ( , , )dV h q q q h q q q h q q q dq dq dq

while the area element with normal in some direction is the cross product of the line element contributions in

the other two directions (following the right hand rule). For example: ˆˆ ˆxdy j dz k dA i or in general:

1 2 3 1 2 3ˆ ˆ ˆ( , , ) ( , , )m m n nmh q q q dq e h q q q dq e dA e

where the sequence ( , , )m n is a right-handed ordering [(1,2,3);(2,3,1);(3,1,2)].

h1 dq1 1e

h2 dq2 2e

h3 dq3 3e 1ˆ ˆn e

1ˆ ˆn e

3ˆ ˆn e

2ˆ ˆn e

(q1,q2,q3)

(q1+dq1,q2+dq2,q3+dq3)

Coordinate Systems

CS 28

Appendix I: The Gradient and Line Integrals

These topics are covered in the Vector Calculus handout, and the current treatment presents just the basic

definitions and techniques. Study the Vector Calculus handouts for the details.

Vector Integration: Conceptually vector integrations are no more difficult than the standard single variable

(1D) integrals studied in freshman Calculus. However they are quite ominous in appearance and often require

more steps to complete their evaluation. In fact, most vector integrals are evaluated by a careful reduction to a

set of 1-D integrals. As an example, the work done on a particle of mass m is computed as it moves along a path

from an initial position ir

to a final position fr

subject to the gravitational attraction of a particle of mass M

located at the origin.

2ˆ

GMmF r

r

The work done on the particle by the gravitational force is given by:

2

ˆf f

i i

r drW F dr GMm

r

where the line element dr

in spherical coordinates is ˆ ˆˆ sindr r r d r d . The coordinate directions are

orthogonal and normalized, and thus ˆ ˆ 1r r , ˆˆ 0r and ˆˆ 0r . The initial and final positions are labeled

by their coordinates as , ,i i ir and , ,f f fr . Substituting, we find:

y

z

x

ir

fr

M

Coordinate Systems

CS 29

2 2ˆ ˆ ˆˆ sin

GMm GMmr dr

r rdr r r d r d

2

1 1f f

i if i

drW F dr GMm GMm

r r r

The integral collapses to a simple integration w.r.t. r because the force field only has a radial component

and the increment of the work done on the particle is the magnitude of the force times the component of the

differential displacement in the direction of the force.

A Guide for Vector Integrations

1. Choose a coordinate system that is appropriate for the problem. For the problem above, the spherical

symmetry of the force field makes a spherical coordinate system centered on the mass M a good candidate.

2. Express all vectors in terms of the coordinate directions for that coordinate system, and compute all inner

(dot) and cross products.

3. If after step 2, a unit vector e in the integrand is not constant w.r.t. the integration variables, replace that

vector by its representation in terms of the constant directions i , j and k . This representation makes the

dependence of the direction e on the integration variables explicit.

4. By this step, the integration has been reduced to either a scalar integration or to a set of scalar integrals

multiplying constant directions. If they are present, the constant directions should be taken outside the integral

leaving a sum of terms each being an integral multiplying a constant direction. This form is a generalization of

the component-wise addition of vectors.

5. Multiple integrals are to be computed as a nested set of single integrations. The techniques to try are:

a. Change of variable: A first guess is the argument of the most complicated function in the integrand.

Be sure to change your limits appropriately and simultaneously!

0 0

( )

( )( ) ' ( )

'

t x t

t x t

dxf x dt f x dxdt

Note that a dummy label t ' is used to represent the integration variable as the integration

variable must be distinct from the limits.

The arguments of functions are dimensionless. Adopting dimensionless variables is a wise and common

practice. Try it!

b. Trig substitutions: Indicated if a. has failed and if the square root of the sum or difference of squares is

Coordinate Systems

CS 30

present.

2

2 2 2 2 2

2 2 2 2

1 ( ) can use sin

tan as 1 tan sec and (tan ) sec

sin as 1 sin cos and (sin ) cos

and sometimes forms like x xb b

xa x d daxa x d da

Be aware that the angle chosen as the new variable is meaningful. Identify on your figure. Interpret

your results in terms of this angle if possible. Square roots are often used to represent distances in physics. If

this is the case, only the positive root is meaningful. Be alert and examine cases. (See page 4 of the

EFieldRing Handout for a trig identities trick.) The hyperbolic functions should be tried if trig

substitution fails (See Basic Integration.).

c. Integration by parts: ( )d u v dv du

u vdx dx dx

or ( )dv d u v du

u vdx dx dx

d. Any tricks presented by your instructor in the course.

e. Integral tables, calculators or Mathematica, but only as a last resort.

6. Reflect on your efforts. Review the techniques that were successful and attempt to identify clues that

would lead you to select them more quickly.

7. Attempt to re-express your results in the language of the problem statement and without any direct

reference to the particular coordinate system that was used. For example: The electric field due to a long

straight uniformly charged wire varies as the inverse of the distance from the wire, and it is directed

perpendicularly away from the wire at any point.

Several sample line integral calculations appear as the first unit in the Tools of the Trade section. Review

them for practical hints.

The Gradient:

Line Elements and Metrics: A general set of coordinates in three dimensions is (q1, q2, q3). The three

directions, 1e , 2e , and 3e , may vary from point to point, but at any point are mutually perpendicular for our

systems. The line element becomes:

1 1 2 3 1 1 2 1 2 3 2 2 3 1 2 3 3 3ˆ ˆ ˆ, , , , , ,q q q dq e q q q dq e q q q dq edr h h h [VCA.1]

Coordinate Systems

CS 31

where the unit vector ie associated with qi is the direction in which the coordinate point moves when the

coordinate qi is given a small positive increment while the other coordinates are held fixed. The functions

hi(q1,q2,q3 ) are the (metric) scale factors for the coordinates and, in particular, the length of dr

squared is:

2 2 2 22 2

2

1 1 2 3 1 2 1 2 3 2 3 1 2 3 3

2

, , , , , ,q q q dq q q q dq q q q dq

ds

h h h

dr dr dr

[VCA.2]

The line elements in our three standard coordinate systems are:

ˆˆ ˆCart i j kdzdr dx dy

ˆˆˆcyl kr dzdr dr r d 2 2

cylr x y [VCA.3]

ˆ ˆˆ sinsphdr dr r r d r d 2 2 2

sphr x y z

Beware: The symbol r is the distance from the axis in cylindrical coordinates and the distance from the origin

in spherical coordinates. This dual usage can cause tremendous difficulties unless you think about what you are

doing. Think about what you are doing at all times. Some authors use the symbols and s to represent

rcylindrical.

The Gradient: The gradient of a scalar function of position is a vector-valued function of position.

[ex: ( ) ( )E r V r

]. If one chose to study the collection of the three partial derivatives for a coordinate

system, the various derivatives could have different dimensions. The gradient is an improvement on partial

derivatives in which the components in the several directions have the same dimensions. A partial derivative is

computed by taking the ratio of the change in the value of the function to the change in one argument when that

argument is varied while the other arguments are held fixed.

0

( , , ) ( , , )y

F F x y y z F x y zLimit

y y

[VCA.4]

The gradient is an associated generalization of the derivative for functions of position in a two, three, or n

dimensional space. The gradient of a scalar function G is a vector function ( )G r

that has, as its component

in a direction, the rate of change of G with respect with respect to distance, not its rate of change with respect to

Coordinate Systems

CS 32

the change of the corresponding coordinate. This distinction between distance and coordinate change is

important, for example, when an angular coordinate is varied. In this case the coordinate change would be d

while the associated distance is r d. Also, all the components of the gradient have the same dimensions, the

dimensions of the function divided by length.

The point is that physics can depend on rates of change with respect to distance. The physics does not depend

on a particular set of coordinates. Physics therefore is more naturally described by gradients that have as their

components the rates of change with respect to distance for each of the independent directions in the coordinate

system. Further the gradient of a scalar field is a vector field while the collection of the three partial derivatives

in spherical coordinates is a mongrel lacking the good transformation properties as it is composed of items that

do not even share the same dimensions (units). [More about this point appears in the linear algebra section.]

More formally, the component of the gradient in the direction e is the rate of change of G with respect to

distance for an infinitesimal displacement in the direction of e , the directional derivative. [That is: G for

d in the direction of interest.] Thus, as x is the displacement in the x direction when x is varied to x + x, the

x component of the gradient is the limit of (G/x) as x approaches zero (= Gx

), the same as the partial

with respect to x). As r is the displacement in the direction when is varied to + , the component of

the gradient is the limit of (G/r) as approaches zero which is 1 Gr

. This result differs from the

partial derivative which is: G

.

A prescription for computing the components of the gradient is the directional derivative given below.

0

0

ˆ( ) ( )ˆ: ( ) ( )

ˆ ˆ( ) ( )

ˆ

n

e s

dr

G r se G rDef e G r G rs

G r dr e e G r

dr e

Lim

Lim

[VCA.5]

For example, the x component of the gradient is x directional derivative:

0 0

ˆ ˆ( ) ( )ˆ( ) ( )ˆ ( ) ( )ˆx

s dr

G r dr i i G rG r s i G ri G r G r

s dr iLim Lim

and the component in cylindrical coordinates is:

Coordinate Systems

CS 33

0 0

ˆ ˆ( ) ( )ˆ( ) ( )ˆ ( ) ( )ˆd dr

G r dr G rG r r d G rG r G r

r d drLim Lim

Review these examples and then derive the forms for all three components of the gradient in each of the three

coordinate systems.

Normal Derivative: The derivative of a function in the direction of the normal at a surface or interface

appears in a variety of problems. It is ˆf f nn

, the direction derivative of ( )f r

in the normal direction.

Note that an alternative definition of the gradient is implicit in the equation

( ) ( ) ( )r r r r rdG G d G G d [VCA.6]

that must be true for arbitrary infinitesimal dr

. The equation states that the inner product of the gradient of G

and the line element must be equal to the total differential of G with respect to its spatial arguments. This

definition also makes explicit the fact that ( )G r

points in the direction in which the function G is increasing

most rapidly with respect to distance and that the magnitude of ( )G r

is the rate of change with respect to

distance in that direction. The area patch formed by a full set of differential displacements perpendicular to is

an equi-G surface patch. As an example, in cylindrical coordinates we would have:

ˆˆˆ ˆˆˆ

r z

r zG G G

dr d dz

dz

dr d dz G r G G k

G dr G r d G

dr r r d dz kdG

As the coordinates can be varied independently, we must equate the coefficients of dr, d and dz individually

with the results that:

; ;r z

G G Gdr d dzdr G dr d G r d dz G dz

[VCA.7]

Exercise: Use Cartesian coordinates to compute ˆr r

where r

is the distance from the origin and r is

the direction away from the origin. Repeat the calculation using spherical coordinates. Motivate the result that

r is direction of most rapid change and that the gradient has magnitude 1. Note that ˆr r

is a vector

Coordinate Systems

CS 34

statement; given the definitions above for r

and r , it is valid in all coordinate systems.

The Gradient Operators

ˆˆ ˆCart i j k

x y z

1 ˆˆˆcyl r k

r r z

[VCA.8]

1 1ˆ ˆˆsinsph r

r r r

The gradient operators have been written with the differential operators rightmost to emphasize that the

derivatives do not operate on the unit vectors or coordinate forms involved in the representation of the gradient

operator itself. These operators are occasionally written with their coordinate directions on the right. This

variant is just a notation, and it is not intended that their evaluation be altered. As represented above, the

derivatives are intended to act on everything to their right.

Exact Forms: A vector field ( )F r

is conservative if it is the gradient of a scalar field ( ( ) ( )F r U r

). The

scalar field ( )U r

is the potential function for the vector field.

( ) ( ) ( ) ( )f f f

i i i

r r r

f i i ir r rU r U r dU U r U dr U r F dr

The form dU F dr

is then an exact differential. A requirement for exactness is that 0F dr for all

paths. This requirement is met if 0F

.

Gradient Summary:

i.) The direction of the gradient of a scalar valued field (function of position) ( )f r

is the direction of

change of the argument for which ( )f r

increases most rapidly.

ii.) The magnitude of the gradient is the rate of change of ( )f r

with respect to the distance by which the

argument is changed in the direction for which ( )f r

increases most rapidly.

In other words: The component of the gradient of a scalar functions in any direction is the rate of change

of that function with respect to the distance by which the argument is incremented in that direction.

Coordinate Systems

CS 35

Alternative: ( ) ( ) ( )r r drr rdf f d f f leading to the fundamental theorem of integral calculus

for scalar functions of position in higher dimension. f f

i i

r r

r rdf f dr

Tools of the Trade

How do we change a vector? How to you change its magnitude? … its direction?

the vector ˆv v v

its magnitude: v v v

its direction: ˆv v

vv v v

Consider the velocity vector for a particle. A small

increment dv

is added to the velocity. That increment

is resolved in to two vectors, one parallel to the initial

velocity and one perpendicular. dv dv dv

As v dv v dv

,

2

v dv v v dv vdv

v v v

. To

finish,

dv dv dv

. First, compute the change in magnitude due to the incremental addition dv

.

12( )

dv v v dv v dvdv d v v dv

v v v v

It is the component of dv

that is parallel to the initial velocity that increases its magnitude. Next, examine the

direction.

3/ 2 3

12

ˆ ˆ2ˆ

v dv vv dv dv vv dv v dvv dv dvdv d

v v v vv v v v v v

As it must be, it is the perpendicular component of dv

that changes the direction. It is, after all, the part that is

in a different direction.

Area Elements: dA = 1 2dr dr

dv

vdv

dv

Coordinate Systems

CS 36

Locally Orthogonal Coordinate Systems for Three Dimensions:

Mathworld http://mathworld.wolfram.com/LaplacesEquation.html

Coordinate

System

Variables Solution Functions

Cartesian

exponential functions, circular functions, hyperbolic

functions

circular

cylindrical

Bessel functions, exponential functions, circular

functions

conical ellipsoidal harmonics, power

ellipsoidal ellipsoidal harmonics

elliptic

cylindrical

Mathieu function, circular functions

oblate

spheroidal

Legendre polynomial, circular functions

parabolic Bessel functions, circular functions

parabolic

cylindrical

parabolic cylinder functions, Bessel functions, circular

functions

paraboloid

al

circular functions

prolate

spheroidal

Legendre polynomial, circular functions

spherical Legendre polynomial, power, circular functions

Laplace's equation can be solved by separation of variables in all 11 coordinate systems that the Helmholtz differential equation

Problems

Coordinate Systems

CS 37

1.) Compute the surface area of a sphere:

,

sphere rA dA

Give the expression for dAr and the integration ranges for and before beginning the integration.

2.) Compute the line integral of E d from ∞ to the point (r,,) where E

is the field of a point charge q

located at the origin. Use the dot products of the pairs of unit vectors to simplify before integrating. The

symbols k and q represent constants.

, , , ,

2ˆ, , 0( )

r r k qr E dr r dr

rV V

3.) A right circular cylinder of radius r is concentric with the z-axis and is bounded by the planes z = 0 and z =

L. Compute the area of one end by using dAz. Compute the total area of the curved surface of the cylinder.

What is the normal direction for this piece? Is it constant? Note that the three surface elements taken

together are a closed surface. In the case that a surface is closed, the outward directed normal is

understood.

4.) The electric field of a uniform line of charge on the z axis can be represented as 2 ˆkr r in cylindrical

coordinates where k and are constants. For this field, compute the integral of E dA

over the surface

described in number 3. Recall that ˆdA n dA

so ˆE dA E n dA

.

ˆ ?E dA E n dA

5.) What is ˆ r in spherical coordinates? …… in cylindrical coordinates?

6.) Prepare a table with expressions for the spherical and cylindrical unit vectors in terms of i , j , and k and

functions of the spherical and cylindrical coordinates. Use the information to compute the integral of the

polar radial direction r with respect to from /2 to . 2 2

ˆ ˆˆ cos sinr d i j d

Compare the

method with one in which r is expressed as: ˆd

d

. In order to finish the evaluation, the difference between

coordinate directions at different points must be computed. Provide sketches of those directions for the

Coordinate Systems

CS 38

distinct coordinate values and the representation of each direction in terms of the Cartesian directions: i , j ,

and k .

7.) Compute the volume of a spherical shell of radius r and thickness dr. First consider sweeping out the volume

by moving each differential area making up the surface of the shell dr in the direction of its normal. Next

compute dVsphere = Vspheree(r+dr) - Vsphere(r) in the limit that dr is small:

( ) ( ) spheresphere sphere sphere

dVV r dr V r dr A dr

dr

8.) For spherical coordinates, sketch the constant coordinate surface = π/4. Prepare a second drawing for the

surface = 3π/4.

9.) Give expressions for the position vector in each of the three coordinate systems. Make sketches showing the

steps from the origin to the point broken up into steps along each coordinate direction. How many steps are

required in each system? Hint: review the tables !

10.) Sketch the three area elements and the volume element in cylindrical coordinates and in spherical

coordinates. Highlight the line element components that define three sides of each volume element. Draw

the volume element first. Each small coordinate side is an area element. The volume element in spherical

should have a small coordinate value corner at (r, , ) and a large coordinate corner at (r + dr, d,

d). Each line element segment should start at (r, , ). What should be true for the cylindrical case?

11.) Use the representations of the cylindrical coordinate directions in terms of the Cartesian set to verify the

expressions for their derivatives w.r.t. .

12.) For spherical coordinates use well-drawn figures to develop the expressions for the partial derivatives of . It is helpful to translate each unit vector back to the origin as was done for the directions in cylindrical

coordinates. Drawn from the origin, the tip of r follows a radius 1 circle of longitude as is varied and a

radius sin circle of latitude as is varied.

Coordinate Systems

CS 39

13.) For Cartesian coordinates, assume that x, y and z depend on time. Compute v

and a

,the first and second

time derivatives of ˆˆ ˆr x i y j z k

. Compute 2v v v

, a handy quantity for representing the kinetic

energy.

Partial Answer: ˆˆ ˆv x i y j z k where dxx dt

14.) For cylindrical coordinates, assume that r, and z depend on time. Compute v

and a

, the first and second

time derivatives of ˆˆr r r z k

. Compute 2v v v

, a handy quantity for representing the kinetic energy.

Partial Answer: ˆˆˆv r r r z k where drr dt

15.) For spherical coordinates, assume that r, and depend on time. Compute v

and a

, the first and second

time derivatives of ˆr r r . Compute 2v v v

, a handy quantity for representing the kinetic energy.

Partial Answer: ˆ ˆˆ sinv r r r r where drr dt

16.) a.) Use the representations of the spherical coordinate directions in terms of the Cartesian unit vectors to

verify the expressions for ˆˆ ˆ

, , andr r

. b.) Continue to find ˆ ˆ ˆ

, , and

.

Partial answers: ˆˆ ˆ ˆˆ, sin cos

rr

17.) Cartesian coordinates are an example of a general locally orthonormal coordinate system. Identify the

hi(q1, q2, q3) dq1 and the ie for Cartesian coordinates. Verify that the area element with normal direction

1e is 1 2 1 2 3 3 1 2 3 2 3( , , ) ( , , )dA h q q q h q q q dq dq .

that the volume element is

1 1 2 3 2 1 2 3 3 1 2 3 1 2 3( , , ) ( , , ) ( , , )dV h q q q h q q q h q q q dq dq dq

and that the unit vector 11

1 1 2 3 1 2 3

1 2 3 1

( , , ) ( , , )ˆ

( , , )r q dq q q r q q q

eh q q q dq

Coordinate Systems

CS 40

18.) Cylindrical coordinates are an example of a general locally orthonormal coordinate system. Identify the

hi(q1, q2, q3) dqi and the 1e for cylindrical coordinates. Verify that the area element with normal

direction 1e is

1 2 1 2 3 3 1 2 3 2 3( , , ) ( , , )dA h q q q h q q q dq dq ,

that the volume element is

1 1 2 3 2 1 2 3 3 1 2 3 1 2 3( , , ) ( , , ) ( , , )dV h q q q h q q q h q q q dq dq dq

and that the unit vector

11

1 1 2 3 1 2 3

1 2 3 1

( , , ) ( , , )ˆ

( , , )r q dq q q r q q q

eh q q q dq

.

19.) Spherical coordinates are an example of a general locally orthonormal coordinate system. Identify the

hi(q1, q2, q3) dqi and the ie for Spherical coordinates. Verify that the area element with normal direction

1e is

1 2 1 2 3 3 1 2 3 2 3( , , ) ( , , )dA h q q q h q q q dq dq ,

that the volume element is

1 1 2 3 2 1 2 3 3 1 2 3 1 2 3( , , ) ( , , ) ( , , )dV h q q q h q q q h q q q dq dq dq

and that the unit vector

11

1 1 2 3 1 2 3

1 2 3 1

( , , ) ( , , )ˆ

( , , )r q dq q q r q q q

eh q q q dq

.

20.) Make a generic sketch of a general locally orthogonal coordinate system. Sketch the three components of

the line element along the sides of a coordinate "cube". Make additional sketches that motivate the

expressions for dA2 and dV.

21.) For each of our three primary coordinate systems, give expressions for the line element dr

and for the

velocity drdtv

. The coordinate directions in each of the systems are orthogonal. Give the expression

for 2v in each system. Re-express your answer using the super-dot notation to represent time derivatives.

2

2;

dr d rr r

dt dt , etc.

Coordinate Systems

CS 41

22.) Compute the metric elements for cylindrical coordinates.

g ij = 3

1

k k

k i j

x x

q q

ANSWER: g11 = 1, g22 = r 2 , g33 = 1, g12 = g21 = g13 = g31 = g32 = g23 = 0

23.) Consider the x-y plane with coordinates q1 = x and q2 = 12

(x + y). Sketch a grid of lines with constant

q1 and constant q2.. Solve for x(q1,q2) and y(q1,q2). Compute g ij = 2

1

k k

k i j

x x

q q

. Show that

22 2

, 1ij i j

i j

g dq dq dx dy

.

ANSWER: g11 = 2, g22 = 2 , g12 = g21 = - 2 ; 1dx dq ; 2 12dy dq dq

Avoid systems that are not locally orthogonal at all costs !

24.) Compute the metric elements for spherical coordinates.

g ij = 3

1

k k

k i j

x x

q q

ANSWER: g11=1, g22= r2 , g33= r

2 sin2, g12 = g21 = g13 = g31 = g32 = g23 = 0

25. Use the representations of the cylindrical coordinate directions in terms of the Cartesian set to verify the

expressions for their derivatives w.r.t. .

26. For spherical coordinates use well-drawn figures to develop the expressions for the partial derivatives of .

It is helpful to translate each unit vector back to the origin as was done for the directions in cylindrical

coordinates. Drawn from the origin, the tip of r follows a radius 1 circle of longitude as is varied and a

radius sin circle of latitude as is varied.

Coordinate Systems

CS 42

27. For Cartesian coordinates, assume that x, y and z depend on time. Compute v

and a

,the first and second

time derivatives of ˆˆ ˆr x i y j z k

. Compute 2v v v

, a handy quantity for representing the kinetic energy.

Partial Answer: ˆˆ ˆv x i y j z k where dxx dt

28. For cylindrical coordinates, assume that r, and z depend on time. Compute v

and a

, the first and second

time derivatives of ˆˆr r r z k

. Compute 2v v v

, a handy quantity for representing the kinetic energy.

Partial Answer: ˆˆˆv r r r z k where drr dt

29. For spherical coordinates, assume that r, and depend on time. Compute v

and a

, the first and second

time derivatives of ˆr r r . Compute 2v v v

, a handy quantity for representing the kinetic energy.

Partial Answer: ˆ ˆˆ sinv r r r r where drr dt

30.

31. For cylindrical coordinates, sketch the constant coordinate surfaces: (a.) r = 1, (b.) = π/4 and (c.) z = -1.

(d.) Describe the coordinate orbit with r = 1 and z = -1. (e.) Describe the r coordinate orbit with = π/4

and z = -1. Describe Prepare a sketch and a prose characterization of the coordinate path (orbit).

32.) Uniform Motion in Polar Coordinates.

Velocity components and coordinate directions

counter vary!

Consider a particle moving at a constant velocity vo in the y -direction. Its acceleration should be zero. The

particle crosses the x axis at t = 0. Its distance from the origin is 2 2 20 or r v t and cos = 0

2 2 20 o

rr v t

. At t

= 0, the velocity is 0 0ˆˆ ˆ0v v i r v

. When the particle’s position line make an angle of with respect to the

x axis, 0 0 0ˆˆ ˆsin cosv v i v r v

. Comparing with the standard results for polar coordinates

ˆ ˆˆ ˆrv r r r v r v and ˆ ˆˆ ˆ2 ra r r r r r a r a

. Using v = vo cos = r , show that:

x

y

v ot

ro

r

x

y

ro

r

r

v

v

Coordinate Systems

CS 43

0 02 2 2

0 0

v r

r v t

. Compute . Based on the expression for r(t), compute r and r. Substitute all these results into

ˆ ˆˆ ˆ2 ra r r r r r a r a . Recall that all these parameters have been evaluated for a particle

traveling at constant velocity. The polar coordinates of the velocity are not constants in time because they must

vary to counter the changes in the coordinate directions as the particle moves from location to location. It is the

overall vector ˆ ˆˆ ˆrv r r r v r v that is to be constant in time.

33.) A vector has the spherical representation ˆˆ10 10r at the point (r,,) = (2,/2,). Give the Cartesian

representation of this vector. (Assume the standard relative orientation of the spherical system relative to

the Cartesian.) Solve the problem two ways. First, replace the spherical coordinate directions by their

general representation in terms of the Cartesian directions. Evaluate the result for = /2 and = . Second,

prepare a careful drawing with the spherical directions sketched at the point (r,,) = (2,/2,). Sketch the

vector ˆˆ10 10r and read its Cartesian components off the drawing.

34.) A vector has the cylindrical representation ˆˆ10 10r at the point (r,z) = (2,/2,). Give the Cartesian

representation of this vector. (Assume the standard relative orientation of the cylindrical system relative to

the Cartesian.)

35.) Referring to transformation matrices

,,

ˆˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆˆ ˆ

ˆˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆˆ ˆ

x r r

y

z zCar cyl sphCart sphCart cyl

i r i i k i r i iF F F

F j r j j k F j r j j F

F F Fk r k k k k r k k

a.) Argue that the transformation matrix cyl,Cart is the transpose of the matrix Cart,cyl.

b.) Give the form of the matrix sph,cyl.

36.) a.) A vector has the spherical representation ˆ ˆˆ10 10 10r at the point (r,,) = (2,/2,). Give the

cylindrical coordinates of that point. Give the cylindrical representation of this vector at that point. (Assume

the standard relative orientation of the spherical system relative to the Cartesian.)

b.) A vector has the spherical representation ˆ ˆˆ10 10 10r at the point (r,,) = (2,/4,). Give the

cylindrical coordinates of that point. Give the cylindrical representation of this vector at that point.

Coordinate Systems

CS 44

37.) Given the Cartesian representations for the spherical coordinate directions.

ˆˆ ˆˆ sin cos sin sin cosr i j k

ˆˆ ˆ ˆcos cos cos sin sini j k

ˆ ˆ ˆsin cosi j

Compute the derivative of with respect to . Note that the result is parallel to the x-y plane. Construct a

direction e that is a linear combination of r and such that has no z-component.

e = a r + b ; ˆˆ 0e k ; ˆ ˆ 1e e

Begin by showing that ˆ ˆe e a2 + b2. Express the derivative of with respect to in terms of r and .

38.) Given that e is a coordinate direction that changes with time as the position change with time. What is ˆ ˆe e

? Show that ˆ

ˆ 0de

edt

. Why must an infinitesimal change in a coordinate direction be perpendicular to that

direction? What feature of the vector would be changed by a small addition parallel to the original vector?

39.) Compute 2L L L

using the spherical coordinate representation. Recall the the differential operators in

the left-most L

are hungry to the right and that they act on the coordinate directions to their right. Recall:

ˆ ˆˆ ˆ ˆ ˆˆ ˆ; cos ; 0; sin cosr r .

40.) Prepare careful drawings of a coordinate “cube” at a point P in spherical coordinates. It should have

diagonally opposite vertices (r, , ) and (r + dr, d, d) . Highlight the three edges that start at the

small coordinate corner (r, , ) and label them as elements of the line element. One would bear the label

ˆdr r for example.

41.) Prepare careful drawings of a coordinate “cube” at a point P in cylindrical coordinates. It should have

diagonally opposite vertices (r, z) and (r + dr, d, zd z). Highlight the three edges that start at the

Coordinate Systems

CS 45

small coordinate corner (r, z) and label them as elements of the line element. One would bear the label

ˆdr r for example.

42.) Prepare careful drawings of a coordinate “cube” at a point P in spherical coordinates. It should have

diagonally opposite vertices (r, , ) and (r + dr, d, d) . Highlight the three edges that start at the

small coordinate corner (r, , ) and label them as coordinate directions. One would bear the label r for

example. Prepare a second drawing with the base of each coordinate direction translated back to the origin

with its direction preserved. Carefully express each direction in terms of i , j , k and functions of the

spherical coordinates.

80.) Prepare careful drawings of a coordinate “cube” at a point P in cylindrical coordinates. It should have

diagonally opposite vertices (r, z) and (r + dr, d, zd z) . Highlight the three edges that start at the

small coordinate corner (r, z) and label them as coordinate directions. One would bear the label r for

example. Prepare a second drawing with the base of each coordinate direction translated back to the origin

with its direction preserved. Carefully express each direction in terms of i , j , k and functions of the

cylindrical coordinates.

81.) A line is is described by the equation x = y = 2 z. Consider the line segment directed along this line from

the plane x = 1 to the plane x = 2. What are the direction cosines of this segment with respect to the

Cartesian coordinate directions?

82.) A line is is described by the equation 3 x = y = 2 z. Consider the line segment directed along this line from

the plane x = 2 to the plane x = 3. What are the direction cosines of this segment with respect to the

Cartesian coordinate directions?

83.) What are the direction cosines of the x direction relative to the spherical coordinate directions at r = 2,

= /4 and = /6?

Coordinate directions are perpendicular to the corresponding constant coordinate surfaces and are directed

in the sense of increasing coordinate value. Hence, if an expression for that coordinate value can be

formulated in terms of the coordinates of a second system system, the gradient of that expression will be in

Coordinate Systems

CS 46

the direction of that coordinate direction, and it will be expressed in terms of the coordinates directions of

the second system. 12 2 2( ) (cos [ ])z

x y z

is in the direction of and is expressed in terms of the

Cartesian directions if the Cartesian representation of the gradient is used.

84.) The direction of the gradient of a scalar is the direction in which the scalar function changes most rapidly

with distance moved. So ( )y

should be in the j direction. In spherical coordinates, y = r sin sin.

Compute the gradient of r sin sin using the spherical representation for the gradient to find the

expression for j in terms of the spherical coordinates and directions. Use analogous methods to find

representations for the other two Cartesian coordinate directions. Explain why your results were the

directions sought without normalizing the gradient to unit magnitude.

85.) The direction of the gradient of a scalar is the direction in which the scalar function changes most rapidly

with distance moved. So ( )y

should be in the j direction. In cylindrical coordinates, y = r sin. Compute

the gradient of r sin using the cylindrical representation for the gradient to find the expression for j in

terms of the cylindrical coordinates and directions. Use analogous methods to find representations for the

other two Cartesian coordinate directions. Explain why your results were the directions sought without

normalizing the gradient to unit magnitude.

86.) The direction of the gradient of a scalar is the direction in which the scalar function changes most rapidly

with distance moved. So 12 2 2( ) (cos [ ])z

x y z

should be in the direction. Compute the

Cartesian gradient to find in terms of the Cartesian coordinate directions. Note that you will need to

normalize the gradient to unit magnitude to ensure that it is a proper direction. Use analogous methods to

find representations for the other two coordinate directions ˆˆ andr .

87.) The direction of the gradient of a scalar is the direction in which the scalar function changes most rapidly

with distance moved. So 12 2( ) (tan [ ])y

x y

should be in the cylindrical direction. Compute the

Cartesian gradient to find in terms of the Cartesian coordinate directions. Note that you will need to

Coordinate Systems

CS 47

normalize the gradient to unit magnitude to ensure that it is a proper direction. Use analogous methods to

find representations for the other two cylindrical coordinate directions ˆˆ andr k .

88.) Show that: 2

0 0

43ˆ ˆ( )( )sin ( )A r B r d d A B

.

Hint: Use ˆˆ ˆˆ sin cos sin sin cosr i j k

A6.) You are to make a sketch displaying spherical coordinates and the small changes of position that follow

from making a small positive increment to each coordinate in turn while holding the other two coordinates

fixed. Each sketch should be at least 3” x 3”. First draw reference Cartesian axis (x,y,z). Next draw a line from

the origin of length 1.5” that makes and angle of (about) 30o and whose projection onto the x-y plane makes

and angle of about 400 with respect to the x axis. Put a small dark dot on the end of the line away from the

origin. Label it P. Sketch a line of longitude through the point P. What is the radius of that arc? Sketch a line of

latitude through P. What is the radius of that arc? Starting at P darken the piece of the line of longitude that

represents increasing to + d. Starting at P darken the piece of the line of latitude that represents increasing

to + d. Using our known relation that the length of an arc that subtends an angle at the center of a circle

has a length equal to the radius of the arc multiplying the angle subtended: s = Rarc, give the lengths of the d

and d arcs. Lightly sketch the latitude arc that joins the points (r,+d,) and (r,+d,d) and the longitude

arc that joins (r,,d) and (r,+d,d). Represent the small patch of area dAr as the product of the lengths

(r,,) to (r,+d,) and (r,,) to (r,,d). The subscript r indicated that the normal direction to this patch of

area is radially away from the origin. Compute 2

0 0 rdA

. If r is increased to r + dr, the patch dAr sweeps

out a volume dV = dr dAr. Compute: 2

0 0 0sin

RV dr d d

.

Make a copy of your sketch as an aid for a later assignment.

A7.) You are to prepare a sketch displaying spherical coordinates and the small changes of position that follow

from making a small positive increment to each coordinate in turn while holding the other two coordinates

fixed. Each sketch should be at least 3” x 3”. First draw reference Cartesian axis (x,y,z). Next draw a line from

the origin of length 1.5” that makes and angle of (about) 30o and whose projection onto the x-y plane makes

and angle of about 400 with respect to the x axis. Put a small dark dot on the end of the line away from the

Coordinate Systems

CS 48

origin. Label it P. Draw a small dark line beginning at P that represents the motion when r is increases to r + dr.

How long is that line? Sketch a line of longitude through the point P. What is the radius of that arc? Sketch a

line of latitude through P. What is the radius of that arc? Starting at P darken the piece of the line of longitude

that represents increasing to + d. Starting at P darken the piece of the line of latitude that represents

increasing to + d. Using our known relation that the length of an arc that subtends an angle at the center

of a circle has a length equal to the radius of the arc multiplying the angle subtended: s = Rarc, give the lengths

of the d and d arcs. Lightly sketch the latitude arc that joins the points (r,+d,) and (r,+d,d) and the

longitude arc that joins (r,,d) and (r,+d,d). Lightly sketch r to r + dr lines at each point. Join the

corners to form a spherical coordinate cube. What is its volume? The radial direction r is the direction that the

point moves when you make a small increase in r while holding the other coordinates fixed. It is the direction

UP on the earth’s surface. How would you define and ? First describe those directions in terms related to

circles of latitude and longitude. Next, interpret them as geographic directions on the earth’s surface. Form the

line element d by multiplying the distance moved for r r + dr by the radial direction and adding the

product of multiplying the distance moved for + d by the direction and adding … . That is:

( , , ) ( , , )d r r dr d d r r . Sketch the line element as a sequence of essentially orthogonal

(perpendicular) small displacements. You will need it when you compute integrals like

sinx y z rF d F dx F dy F dz F dr F rd F r d .

RELAX: They are easier than they look if you can visualize the sketches assigned above.

Integrate dV to show that the volume in a spherical shell with inner radius R and outer radius R +dr is

approximately 4R2 dr for dr << R. Compare with dVdrdV dr where V = 4/3 r

3.

A8.) Generate problem analogous to A6 and A7 for cylindrical coordinates and solved them. Replace

geographic with prose explanations.

A9.) Analytic development of the coordinate directions for locally orthonormal systems.

1 1 2 2 3 3 1 2 3( , , ) ( , , )dr d r q dq q dq q dq r q q q

1 1 2 2 3 31 2 3 1 1 2 3 2 1 2 3 3ˆ ˆ ˆ( , , ) ( , , ) ( , , )dr d h q q q dq e h q q q dq e h q q q dq e

Note that a differential change of position is often represented as .

Begin with the Cartesian case: ; . This system has the unique

additional property that its coordinate directions are global constants – important because we are to compute

Coordinate Systems

CS 49

derivatives. Suppose we want to develop these relations for cylindrical coordinates. Express the Cartesian

coordinates using the cylindrical ones.

q1,q2,q3 r,,z

Use , , tofindthemetricfactorsandcoordinatedirections.

1 ⟹ 1, ,

a. Repeatfor and andidentifyh,hz, .

b. Repeatforallthreeofthesphericalcoordinates.

A10. Followingtheprocessabovewefindthecylindricalcoordinatedirections.

,

Nextweuse: ∙ , , ∙ ∙ , .

RepeatfortheremainingCartesiandirections.Wenowhavetransformationstoexpresscoordinate

directionsintermsoftheCartesiandirectionsandtorepresenttheCartesiandirectionsintermsofthose

forthenewcoordinatesystem.

A11. RepeatA10forthesphericalcoordinatedirections.

A12. ReviewA10.Developmatricestotransformbetweencylindricalandsphericalcoordinate

directions.Stateinproseformthemethodtodevelopthetransformationsbetweenthecoordinate

directioncomponentsofanytwolocallyorthonormalcoordinatesystems.

References:

1. K. F. Riley, M. P. Hobson and S. J. Bence, Mathematical Methods for Physics and Engineering, 2nd Ed.,

Cambridge, Cambridge UK (2002).

2. Mary L. Boas, Mathematical Methods in the Physical Sciences, 2nd Edition, chapter 3, John Wiley & Sons

(1983).

Coordinate Systems

CS 50

3. Donald A. McQuarrie, Mathematical Methods for Scientists and Engineers, University Science Books,

Sausalito, CA (2003).

4. The Wolfram web sites: mathworld.wolfram.com/ and scienceworld.wolfram.com/

Related Problems:

MD:31.) The volume of a parallelepiped with sides , ,A B C

is A B C

. Show that this volume is can be

computed as the determinant x y z

x y z

x y z

A A A

B B B

C C C

.

MD:32.) A general set of 3D coordinates are related to the Cartesian set by the transformation equations

x(q1,q2,q3), y(q1,q2,q3), and z(q1,q2,q3). Small positive changes in each of the coordinates generate three small

displacements characteristic of the general coordinates (assumed to be in 'RH' order):

1 1 1 11 1 1

ˆˆ ˆq

yx zq q qdr dq i dq j dq k

,

2 2 2 22 2 2

ˆˆ ˆq

yx zq q qdr dq i dq j dq k

,

3 3 3 33 3 3

ˆˆ ˆq

yx zq q qdr dq i dq j dq k

and 1 2 3

0q q qdr dr dr

. Show that the volume element in the general system can be represented as:

1 1 1

2 2 2 1 2 3

3 3 3

1 1 1 1 1 1

2 2 2 2 2 2

3 3 3 3 3 3

y yx z x zq q q q q q

y yx z x zq q q q q q

y yx z x zq q q q q q

dq dq dq

dV dq dq dq dq dq dq

dq dq dq

where

1 1 1

2 2 2

3 3 3

yx zq q q

yx zq q q

yx zq q q

is the determinant of the Jacobian matrix for the transformation.

MD:33.) Compute

yx zr r r

yx z

yx z

dV dr d d

. It is helpful to factor r2 sin out of each term in the

expansion of the determinant. Refer to the previous problem.

Coordinate Systems

CS 51