Vector Calculus - Mathematical Methods of Physics | Physics for IIT JAM, UGC - NET, CSIR NET PDF Download

Introduction

Imagine you’re exploring the invisible forces of nature—how does electricity move through a circuit, how do planets follow their curved paths, or how do fluids flow effortlessly through pipes?

 The answer lies in vector calculus, the language of physics that reveals the hidden patterns in the universe. 

Vector Calculus equips you with tools like the gradient to find steepest climbs, the divergence to measure spreading flows, and the curl to detect swirling motions. Whether you're decoding Maxwell's equations in electromagnetism or analyzing fluid dynamics, vector calculus is your key to unlocking the mysteries of the physical world!

Gradient of a Vector

Suppose that we have a function of three variables-say, V (x, y, z) in a

This tells us how V changes when we alter all three variables by the infinitesimal amounts dx, dy, dz. Notice that we do not require an infinite number of derivatives-three will suffice:

 the partial derivatives along each of the three coordinate directions. 

Thus

where  is the gradient of V.

 is a vector quantity, with three components.

Geometrical Interpretation of the Gradient

 Like any vector, the gradient has magnitude and direction. To determine its geometrical meaning, let’s rewrite:

where is the angle between and

Now, if we fix the magnitude  and search around in various directions (that is, vary ), the maximum change in V evidently occurs when = 0 (for then  cos = 1). That is, for a fixed distance , dT is greatest when one move in the same direction as .

Thus, the gradient  points in the direction of maximum increase of the function V.

Moreover, the magnitude  gives the slope (rate of increase) along this maximal direction.

Gradient in Spherical polar coordinates

Gradient in cylindrical coordinates

Example 1: Find the gradient of a scalar function of position V where Calculate the magnitude of gradient at point

Example 2: Find the unit vector normal to the curve y = x2 at the point (2, 4, 1).

Solution: The equation of curve in the form of surface is given by  

A constant scalar function V on the surface is given by V (x, y, z) = x² - y 

Taking the gradient  

 The value of the gradient at point (2, 4, 1),  

The unit vector, as required        

Example 3:  In electrostatic field problems, the electric field is given by where V is the scalar field potential. If  in spherical coordinates, then find .

The Operator

The gradient has the formal appearance of a vector “multiplying” a scalar V:

The term in parentheses is called “del”:     

  We should say that is a vector operator that acts upon V, not a vector that multiplies V.

There are three ways the operator can act:

• On a scalar function V:  (the gradient);
• On a vector function , via the dot product: (the divergence);
•  On a vector function, via the cross product: (the curl).

Divergence of a Vector

If  is a vector point function then   is called Divergence of .

where  are the functions of x, y, z

Note:

1. Divergence of a vector is scalar.
2. Physically Divergence measures (outflow - inflow)
3. A vector whose divergence is zero then it is said to be divergence free vector (or) solenoid vector i.e. outflow = inflow = constant

Geometrical Interpretation

  is a measure of how much the vector  spreads out (diverges) from the point in question. For example, the vector function in figure (a) has a large (positive) divergence (if the arrows pointed in, it would be a large negative divergence), the function in       figure (b) has zero divergence, and the function in figure (c) again has a positive divergence.

Divergence in Spherical polar coordinates  

Divergence in cylindrical coordinates 

The Curl  of a Vector

 From the definition of  we construct the curl

Geometrical Interpretation

 is a measure of how much the vector   “curls around” the point in question. Figure shown below have a substantial curl, pointing in the z-direction, as the natural right-hand rule would suggest.  

Curl in Spherical polar coordinates 

Curl in cylindrical coordinates 

Angle between Two Surfaces

Let ϕ₁(x,y,z) = C, ϕ₂(x,y,z) = C be given equations of two level surfaces and angle between these two surfaces are given as θ then cos θ

Note:

The angle between two surfaces is nothing but the angle between their normal.

 then they are said to be orthogonal surfaces.

 Example 4:  The angle between the two surfaces x² + y²+ z² = 9 and z = x² + y² − 3 at the point (2, −1, 2) is

Solution:

Example 5:  Find the curl of the vector

Directional Derivatives of a Scalar Function 

The directional derivative of a scalar function ϕ (x, y, z) in the direction of a vector

given as

ve then it is in the opposite direction.

Example 6: The Directional derivative of f(x, y, z) = x²yz + 4xz² at (1, −2, −1) along (2i – j − 2k) is
Solution:

Directional Derivative =

At (1, −2, −1) we have

Example 7: The values of a, b, c so that the vector,

is irrotational.

Solution: 

Given, that vector V is irrotational 

⇒ c = −1, a = 4, b = 2

∴ a = 4, b = 2, c = −1

Second Derivatives   

The gradient, the divergence, and the curl are the only first derivatives we can make with  by applying  twice we can construct five species of second derivatives. The gradient is a vector, so we can take the divergence and curl of it:

•  Divergence of gradient: 

This object, which we write  for short, is called the Laplacian of V. Notice that the Laplacian of a scalar V is a scalar.

Laplacian in Spherical polar coordinates

Laplacian in cylindrical coordinates 

Occasionally, we shall speak of the Laplacian of a vector, . By this we mean a vector quantity whose x-component is the Laplacian of A_x, and so on:

•  Curl of gradient: 

The divergence  is a scalar-all we can do is taking its gradient.

The curl of a gradient is always zero:

• Gradient of divergence: 

The curl  is a vector, so we can take its divergence and curl.

Notice that  is not the same as the Laplacian of a vector:

•  Divergence of curl: 

The divergence of a curl, like the curl of a gradient, is always zero: 

• Curl of curl:

As you can check from the definition of

So curl-of-curl gives nothing new; the first term is just number (3) and the second is the Laplacian (of a vector).