1. scalar fields and vector fields the simplest possible...

3FI4: Theory and Applications in Electromagnetics

Lecture 2: Vector Analysis: Differential Calculus page 1 September 12, 2000

1. Scalar Fields and Vector Fields

The simplest possible physical field is the scalar field. Itrepresents a function depending on the position in space (andtime). A scalar field is characterized at each point in space(and time) by a single number.

Examples of scalar fields: temperature, gravitationalpotential, electrostatic potential (voltage), etc.



Fig. 1. Equipotential plot of a scalar field



A vector field is a vectorial quantity that depends on the positionin space (and time).

Fig. 2. Arrow plot of a vector field



2. Gradient

The gradient of a scalar field is a vector whose magnitudeis equal to the maximum rate of change of the field andwhose direction is equal to the direction of the fastestincrease.

Assume a scalar field ( , , )x y zΦAn infinitesimal displacement along the x-axis dx brings usto a slightly different scalar value ( , , )x dx y zΦ +

ˆ( , , ) ( , , )xd x dx y z x y z dl xdxΦ = Φ + − Φ =ˆ( , , ) ( , , )yd x y dy z x y z dl ydyΦ = Φ + − Φ =ˆ( , , ) ( , , )zd x y z dz x y z dl zdzΦ = Φ + − Φ =



An infinitesimal directional displacement

ˆ ˆ ˆdl xdx ydy zdz= + +will invoke all three changes simultaneously:

x y zd d d dΦ = Φ + Φ + Φ

d dx dy dzx y z

∂Φ ∂Φ ∂ΦΦ = + +∂ ∂ ∂

( )gradient

ˆ ˆ ˆ ˆˆ ˆ

dl

d x y z xdx ydy zdzx y z

∂Φ ∂Φ ∂ΦΦ = + + ⋅ + + ∂ ∂ ∂

Lookslike adotproduct



The directional derivative

uu

∂Φ∂

is a vector showing the direction of spatial displacement andthe corresponding rate of change of the scalar function.

In orthogonal coordinate system, the gradient is a vectorialsum of the directional derivatives of the scalar field alongthe three unit vectors of the system

ˆ ˆ ˆgrad x y zx y z

∂Φ ∂Φ ∂ΦΦ ≡ ∇Φ = + +∂ ∂ ∂

In rectangular coordinates:



Equipotential surface is the geometrical place of all points atwhich the scalar field has the same value:

( , , )x y z constΦ =In 2-D problems, the surface would collapse into a line.

x

y

1Φ =

2Φ =

4Φ =

3Φ =∇Φ

xx

∂Φ∂

yy

∂Φ∂

dτ ˆ 0ττ

∂Φ =∂



The directional derivative along the tangent of anequipotential line is zero.

The directional derivative along the normal of anequipotential line has maximum magnitude: this is thegradient.

max

|u

∂Φ∇Φ =∂

The directional derivative in any direction has amagnitude determined by the projection of the gradientonto its direction:

ˆ | | cosuu

θ∂Φ = ∇Φ ⋅ = ∇Φ∂



Gradient in cylindrical coordinate system:

1ˆ ˆ ˆ( , , )r z r z

r r zϕ ϕ

ϕ∂Φ ∂Φ ∂Φ∇Φ = + +∂ ∂ ∂

Note: Whenever an angular increment is involved, it has tobe converted to a linear element, e.g.,

rd dlϕϕ =Gradient in spherical coordinate system:

1 1ˆˆ ˆ( , , )sin

R RR R R

θ ϕ θ ϕθ θ ϕ

∂Φ ∂Φ ∂Φ∇Φ = + +∂ ∂ ∂

x

y

z

ϕ

θ R

sinr R θ=

Rdθ

rdϕ



Illustration of linear elements corresponding to angularincrements in spherical coordinate system



3. Divergence

To estimate and quantify vector fields, it is often useful tomeasure their flow (or net outflow). Such a measure isthe vector field flux. The flux is the net normal flowthrough a surface

cosS S

F ds F dsθΨ = ⋅ =∫∫ ∫∫

ˆds nds=Differential surface area (surface element):



ds

nF



The flux over a closed surface is a measure of the field’ssources in the enclosed volume

S

F dsΨ = ⋅∫∫

0Ψ =0Ψ > 0Ψ <

source sink



The divergence is the net outward flux of a vector field perunit volume (as the volume shrinks to zero):

0div lim VSV

F ds

F FV→

⋅≡ ∇ ⋅ =

∫∫

In rectangular coordinate system:

( , , )( , , ) ( , , )( , , ) yx z

F x y zF x y z F x y zF x y x

x y z

∂∂ ∂∇ ⋅ = + +∂ ∂ ∂

The divergence is a scalar quantity!



x

y

z

x

y

z

0 0 0ˆ ( , , )2xx

xF x y z+

0 0 0ˆ ( , , )2xx

xF x y z− −

0 0 0ˆ ( , , )2xy

yF x y z+0 0 0ˆ ( , , )2xy

yF x y z− −

0 0 0ˆ ( , , )2xz

zF x y z +

0 0 0ˆ ( , , )2xz

zF x y z− −

Divergence in rectangular coordinates illustrated

The differential volume (volume element) is: 0V x y z= →



In general orthogonal coordinate system:

23 1 13 2 12 3

1 2 3

2 3 1 1 3 2 1 2 31 2 3 1 2 3

( , , )

1( ) ( ) ( )

S F S F S FV

F u u u

h h F h h F h h Fh h h u u u

∇ ⋅ =

∂ ∂ ∂ = + + ∂ ∂ ∂

This formula is not scary: it follows directly from thedefinition of the divergence operator

1 2 3, ,h h h are the metric coefficients of the respective OCS,

e.g., in CCS: 1

2

3

: 1:: 1

r hh r

z hϕ

===



Divergence in cylindrical coordinates:

1 1( , , ) ( ) z

r

F FF r z rF

r r r zϕϕ

ϕ∂∂ ∂∇ ⋅ = + +

∂ ∂ ∂

Divergence in spherical coordinates:

22

( , , )

1 1 1( ) ( sin )

sin sinR

F R

FR F F

R R RRϕ

θ

θ ϕ

θθ θ θ ϕ

∇ ⋅ =∂∂ ∂= + +

∂ ∂ ∂



4. The Divergence Theorem (Gauss Theorem)

The outward flux of a vector field over a closed surface Sis equal to the volume integral of this field’s divergenceover the volume confined by S.

[ ]VS V

F ds Fdv⋅ = ∇ ⋅∫∫ ∫∫∫

Adding up the individual outward fluxof all sub-cells results in a net outwardflux over the surface of the wholevolume.



5. Curl (rotor)

The flux (or the divergence) is not enough to describe avector field. It is descriptive only with regard to thecomponents of the field normal to a given surface. Thefield components tangential to a surface (or a contour of asurface) are important, too. Thus, a vector field is alsocharacterized by its circulation.

C

C F dl= ⋅∫

[ ]SCS

dln



The curl of a vector field is the circulation per unit surface (asthe contour and the surface shrink to zero):

0ˆcurl rot lim CS

F dl

F F F nS→

⋅≡ = ∇ × =

∫

The curl is a vector quantity!

0C > 0C <0C =



The curl vector has three orthogonal components in anorthogonal coordinate system. In the RCS, the circulation inthe x-y plane defines the z-component of the curl; thecirculation in the y-z plane defines the x-component; and thecirculation in the x-z plane defines the y-component of thecurl.

z

xy

0 0 0ˆ ( , , )2xy

xF x y z−

0 0 0ˆ ( , , )2xy

xF x y z− +

0 0 0ˆ ( , , )2yx

yF x y z+0 0 0ˆ ( , , )2yx

yF x y z− −

The z-component:



(curl ) y xz

F FF

x y

∂ ∂= −∂ ∂

In an analogous manner:

(curl ) x zy

F FF

z x

∂ ∂= −∂ ∂

(curl ) yzx

FFF

y z

∂∂= −∂ ∂

ˆ ˆ ˆcurl y yz x z xF FF F F F

F x y zy z z x x y

∂ ∂ ∂ ∂ ∂ ∂ = − + − + − ∂ ∂ ∂ ∂ ∂ ∂

ˆ ˆ ˆ

curl

x y z

x y z

Fx y z

F F F

∂ ∂ ∂=∂ ∂ ∂



Since the del operator can be written in vector form as:

ˆ ˆ ˆx y zx y z

∂ ∂ ∂∇ = + +∂ ∂ ∂

the curl can be written as:

curl F F= ∇ ×In any orthogonal coordinate system:

1 1 2 2 3 3

1 2 3 1 2 3

1 1 2 2 3 3

ˆ ˆ ˆ

1

h u h u h u

Fh h h u u u

h F h F h F

∂ ∂ ∂∇ × =∂ ∂ ∂



6. Stokes’ Theorem

The net flux of the curl of a vector over any open surface Sis equal to the line integral of the vector along the closedcontour C enclosing the surface.

[ ]

( )SC S

F dl F ds⋅ = ∇ × ⋅∫ ∫∫

Adding up the individual circulationsof all sub-cells results in a netcirculation over the contour of thewhole surface.



7. Second-order differential operators

Some important identities

0∇ × ∇Φ =

( ) 0F∇ ⋅ ∇ × =

proven by Stokes’ theorem

proven by Gauss’ theorem

2( )F F F∇ × ∇ × = ∇ ∇ ⋅ − ∇

2 2 2 2ˆ ˆ ˆ( ) ( ) ( )x y zF x F y F z F∇ = ∇ + ∇ + ∇In rectangular coordinates:

2 2 22

2 2 2( )x y z

∂ Φ ∂ Φ ∂ Φ∇ Φ = ∇ ⋅ ∇Φ = + +∂ ∂ ∂

1. scalar fields and vector fields the simplest possible...

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