scalars, vectors and fields

SCALARS, VECTORS AND FIELDS

SHAHBAZ AHMED ALVI

1. Scalars

Scalars are those quantities which can be defined by a single number. It justtells you how much of something there is. For example, if someone says thatthe temperature is 32 Co then it means that something is 32 units hotter on atemperature scale or if you say something is moving at a speed of 45 km/h thenit means that the object is covering 45 km in an hour, regardless of its direction.Other examples of scalar quantities are, energy, work done, electrical resistanceetc. A scalar quantity is only represented by a letter or a symbol.

2. Vectors

Quantities categorized as vectors have a magnitude along with a direction. In3 dimensions you need 3 numbers to define a vector completely. These 3 numbersare called components of the vector in each of the three directions. A vector in 3Dis written as,

~A = axi+ ay j + azk

Here, ax, ay and az are called the components of ~A in the direction of unit vectors

i, j and k. A unit vector is a vector which has a magnitude of 1 and a certaindirection. i is the unit vector which points in the direction of x-axis. Similarly jand k are unit vectors in the direction of y and z axes respectively. Examples ofvector quantities include velocity, acceleration, current density, area etc.

3. Vector Products

A vector can be multiplied in three ways.

3.1. Product with a scalar. When a vector is multiplied with a scalar, you geta new vector whose components, and magnitude, are greater than the originalvector.

~A′ = b ~A = baxi+ bay j + bazk

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2 SHAHBAZ AHMED ALVI

Figure 1. A 3 dimensional vector ~A = axi + ay j + azk and itscomponents.

3.2. Product with a vector: Dot Product. A dot product is when a vector ismultiplied to another vector and the result is a scalar. A dot product between ~Aand ~B is defined as,

~A. ~B = | ~A|| ~B| cos θ

By this definition, dot product between like unit vectors is equal to one and unlikeunit vectors have zero dot product. The dot product can also be defined anotherway,

~A. ~B = (axi+ay j+azk).(bxi+by j+bzk) = axbxi.i+axby i.j+axbz i.k+ ....+azbzk.k

Since dot product between unlike unit vectors is zero, the only terms which survivein this sum are with like unit vectors i.e.

~A. ~B = axbxi.i+ ayby j.j + azbzk.k = axbx + ayby + azbz

It is also called a scalar product.

3.3. Product with a vector: Cross Product. A cross product is when a vectoris multiplied to another vector and the result is a vector. A cross product between~A and ~B is defined as,

~A× ~B = | ~A|| ~B| sin θn

Here, n is unit vector which is perpendicular to both ~A and ~B. Defined this way,a cross product between like unit vectors is zero (sin 0 = 0) and cross productbetween unlike unit vectors is equal to the unit vector perpendicular to them (e.g.

SCALARS, VECTORS AND FIELDS 3

i× j = k). A cross product is also defined as determined,

~A× ~B = DET

i j kax ay azbx by bz

It is also called vector product.

4. Fields

A Field is basically a space (a volume or a section of space) where at every pointsome quantity is defined (means some quantity has some value). If this quantityis a scalar, the field is called a Scalar field. If the quantity is a vector the fieldis called a Vector field. So if you have temperature defined at every point in, forexample, a room then the room represents a temperature scalar field. Similarly airmolecules flowing in a pipe have velocity vector pointing in the direction of flow.If you plot the velocity vectors of molecules in the pipe, you got yourself a velocityvector field within the pipe.

4.1. Plotting a field. Plotting a scalar field is simple. You have a scalar function,which looks like any other mathematical function, and you plot that function atevery point in the given space. For example, the function T(x,y,z) = x2y + y2z canrepresent the temperature field in a volume.

A vector field is plotted is a similar way but at every point, in addition to anumber, there is a vector with a direction. A vector function looks like any othervector. For example, if you were to plot a vector field of the vector ~V = 4i+ 5j, itwould look like fig. 2. Everywhere in the space the vector has fixed direction anda fixed magnitude (length)! Which is pretty boring. Things get more interesting(and admittedly attractive) if the components of the field are not fixed but are

variables i.e. ~V = xi or ~V = xi+yj. These vector fields will look like fig. ?? whenplotted: And you can plot a lot of interesting vector fields.

5. Directional Derivative

5.1. Partial Derivative. You have studied Calculus in your previous Mathemat-ics classes and are equipped with the idea of differentiation. But functions thatyou used to differentiate depended only on one variable (i.e. f(x)). Many functionsin Mathematics and Physics depend on more than one variable (i.e. f(x,y,z)). Whendealing with a “multivariable” function you use “Partial derivative” instead of the“ordinary or normal”derivative. A partial derivative is represented by ∂ symbol.The idea behind a partial derivative is to change only one variable at a time whilekeeping all the other variables constant. So if you are differentiating with respectto x, you will keep y and z constant. For example, lets try partially differentiating


(a) A visualization of a vector field.

(b) Velocity field of air flowing through ashaft with a rotor fan.

(c) This is the temperature field of the ob-serveable Universe which was studied by thelatest PLANCK satelite

(d) A scalar field map of air pressure overNorth America. Red shades show higherpressure and purple shades indicate lowpressure.


Figure 2. A constant vector field ~V = 4i + 5j. Everywherethe vector of the field has the same direction and same magnitude(√

42 + 52).

the function f(x,y,z) = xy2 + y2z + z2x.

∂f(x,y,z)∂x

=∂(xy2 + y2z + z2x)

∂x=∂(xy2)

∂x+∂(y2z)

∂x+∂(z2x)

∂x= y2 + 0 + z2

Similarly,

∂f(x,y,z)∂y

= 2xy + 2yz + 0

∂f(x,y,z)∂z

= 0 + y2 + 2zx

In Physics if a certain quantity (like temperature or velocity) depends on morethan one variable then we study it by changing only one variable at one timewhile keeping others constant. This is exactly what a partial derivative does;differentiates one variable at a time and keeps others constant.

5.2. Directional Derivative. The following function is called a Directional de-rivative:

~∇ =∂

∂xi+

∂

∂yj +

∂

∂zk

Just like that, alone, this function has no meaning! It just consists of partialderivatives, we call it a “operator”. This operator gets its meaning when it operateson a scalar or a vector function. Just like any other vector you can multiply it


(a) Vector field defined by ~V = xi. The direction is towards x axis at all points and the

magnitude, |~V | = x (length of the arrow) increases as you move towards larger values ofx.

(b) Vector field defined by ~V = xi+ yj. The direction and magnitude of the field variesin space in the xy plane.

with a scalar or take a dot or a cross product with a vector and each one has adifferent meaning. Suppose a vector field ~V and a scalar field T(x,y,z).


5.3. GRAD(T) = ~∇T(x,y,z). This will give you the “Gradient” of the scalar fieldand the gradient of the field, at a certain point in the field, tells you the directionin which the “rate of change of field (with respect to distance)” is maximum. Ifwe calculate the gradient of the above field we get,

~∇T(x,y,z) =∂T(x,y,z)∂x

i+∂T(x,y,z)∂y

j +∂T(x,y,z)∂z

k

~∇T(x,y,z) = (y2 + z2)i+ (2xy + 2yz)j + (y2 + 2zx)k

And at any given point, (x, y, z), ~∇T(x,y,z) will give the direction in which thetemperature has the maximum rate of change.

5.4. DIV(~V ) = ~∇.~V . This is called the “Divergence” of the vector field ~V . Thedivergence of a field tells you how much the field is diverging at a point or how muchis being “emitted or is coming out” at a certain point. For example, the pictureshows the electric field of a point charge for which the divergence is maximum. Ifthe field is coming out at a point the divergence is positive and if it is going in thenit is negative. The former is called the source of the field and the later is calledthe sink. Like any other dot product, divergence of a field is a scalar. Assume avector field, ~V = xy2i+ y2zj+ z2xk. The divergence of this field can be evaluatedas:

~∇.~V = (∂

∂xi+

∂

∂yj +

∂

∂zk).(xy2i+ y2zj + z2xk)

~∇.~V =∂(xy2)

∂x+∂(y2z)

∂y+∂(z2x)

∂z

= y2 + 2yz + 2zx

And this will give you the amount of field lines coming-out or going-in at a certainpoint in the space of the field. Its hard to imagine but there are some field which

Figure 3. Electric field emitting from a positive charge which actsas the “source” of the electric field while the negative charge acts asthe “sink”.


are divergenceless. Means that are not ”coming out” or ”going in” at any point.One such example is the magnetic field. For instance, look at the magnetic fieldof a bar magnet, how the field lines start at the north pole travel towards thesouth and then through the magnet and end at North again. The field lines areclearly not ending anywhere. Such a field, where field lines form closed loops, isone example of a divergenceless field.

Figure 4. Magnetic field lines of a bar magnet form closed loops.

5.5. CURL(~V ) = ~∇ × ~V . And this is the “Curl” of ~V and the curl tells you ifthe field lines have a rotation at a certain point in space or if the field lines rotate.Since it is a cross product curl of a vector field is a vector. If you rotate yourfingers in the direction of the field lines your thumb will point in the direction ofthe curl vector. Lets evaluate the curl of the same vector field.

~∇× ~V = (∂

∂xi+

∂

∂yj +

∂

∂zk)× (xy2i+ y2zj + z2xk)

~∇× ~V = DET

i j k∂∂x

∂∂y

∂∂z

xy2 y2z z2x

And this is the direction of the curl vector at the point (x, y, z). I told you

that field lines which for closed loops have zero divergence but they such lineshave maximum curl. In normal circumstances Electric field lines do not have acurl i.e. ~∇ × ~E = 0. But there is one situation in which ~∇ × ~E 6= 0 and that isin case of Faradays law. Electric field which is created as a result of alternatingmagnetic field form closed loops and thus have a non-zero curl. In the form ofVector calculus Faraday is written as,

~∇× ~E = −∂~B

∂t


Figure 5. Changing magnetic field lines through a loop causes cir-cular electric field lines which causes current in the loop.

scalars, vectors and fields

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