Unit 6

Vector Differential Calculus

6.1      Velocity vector, acceleration vector

Vector function-

Suppose be a function of a scalar variable t, then-

Here vector varies corresponding to the variation of a scalar variable t that its length and direction be known as value of t is given.

Any vector  can be expressed as-

Here , , are the scalar functions of t.

Differentiation of a vector-

We can denote it as-

Similarly is the second order derivative of

Note- gives the velocity and gives acceleration.

Note-

1. Velocity =

2. Acceleration =

The derivative of the velocity vector V(t) is called acceleration A(t), and it is given as-

Rules for differentiation-

1.

2.

3.

4.

5.

Example-1: A particle moves along the curve , here ‘t’  is the time. Find its velocity and acceleration at t = 2.

Sol. Here we have-

Then, velocity

Velocity at t = 2,

=

Acceleration =

Acceleration at t = 2,

Example-2: If and then find-

1.

2.

Sol. 1. We know that-

2.

Example-3: A particle is moving along the curve x = 4 cos t, y = 4 sin t, z = 6t. Then find the velocity and acceleration at time t = 0 and t = π/2.

And find the magnitudes of the velocity and acceleration at time t.

Sol. Suppose

Now,

At t = 0	4
At t = π/2	-4
At t = 0	|v|=
At t = π/2	|v|=

Again acceleration-

Now-

At t = 0
At t = π/2
At t = 0	|a|=
At t = π/2	|a|=

Scalar point function-

If for each point P of a region R, there corresponds a scalar denoted by f(P), in that case f is called scalar point function of the region R.

Note-

Scalar field- this is a region in space such that for every point P in this region, the scalar function ‘f’ associates a scalar f(P).

Vector point function-

If for each point P of a region R, then there corresponds a vector then is called a vector point function for the region R.

Vector field-

Vector filed is a reason in space such that with every point P in the region, the vector function associates a vector (P).

Note-

Del operator-

The del operated is defined as-

Example: show that  where

Sol. Here it is given-

=

Therefore-

Note-

Hence proved

Key takeaways-

1. Any vector  can be expressed as-

Here , , are the scalar functions of t.

1. Velocity =

2. Acceleration =

3. Scalar field- this is a region in space such that for every point P in this region, the scalar function ‘f’ associates a scalar f(P).

4. Vector field-

Vector filed is a reason in space such that with every point P in the region, the vector function associates a vector (P).

6.2      6.2. Tangential and normal component of acceleration

Tangential and normal accelerations-

It is very important to understand that the magnitude of acceleration is not always the rate of change of |V|.

Suppose be the vector-valued function which denotes the position of any object as a function of time.

Then

Then the tangential and normal component of acceleration are given as below-

Example: A object move in the path where t is the time in seconds and distance is measured in feets.

Then find and as functions of t.

Sol.

We know that-

And

Now we will use-

And now-

Key takeaways-

6.3      Vector differential operator, gradient, Divergence and curl

The vector differential operator is written as-

Gradient

Suppose f(x, y, z) be the scalar function and it is continuously differentiable then the vector-

Is called gradient of f and we can write is as grad f.

So that-

Here is a vector which has three components

Properties of gradient-

Property-1:

Proof:

First we will take left hand side

L.H.S =

=

=

=

Now taking R.H.S,

R.H.S. =

=

=

Here-                     L.H.S. = R.H.S.

Hence proved.

Property-2: Gradient of a constant (

Proof:

Suppose

Then

We know that the gradient-

= 0

Property-3: Gradient of the sum and difference of two functions-

If f and g are two scalar point functions, then

Proof:   

L.H.S                                

Hence proved

Property-4: Gradient of the product of two functions

If f and g are two scalar point functions, then

Proof:

So that-

Hence proved.

Property-5: Gradient of the quotient of two functions-

If f and g are two scalar point functions, then-

Proof:

So that-

Example-1: If , then show that

1.

2.

Sol.

Suppose   and

Now taking L.H.S,

Which is

Hence proved.

2.

So that

Example: If then find grad f at the point (1,-2,-1).

Sol.

Now grad f at (1 , -2, -1) will be-       

Example: If then prove that grad u , grad v and grad w are coplanar.

Sol.

Here-                     

Now-

Apply

Which becomes zero.

So that we can say that grad u, grad v and grad w are coplanar vectors.

Divergence (Definition)-

Suppose is a given continuous differentiable vector function then the divergence of this function can be defined as-

Curl (Definition)-

Curl of a vector function can be defined as-

Note- Irrotational vector-

If then the vector is said to be irrotational.

Vector identities:

Identity-1:  grad uv = u grad v + v grad u

Proof: 

So that

Grad(uv) = u grad v + v grad u

Identity-2: 

Proof:

Interchanging , we get-

We get by using above equations-

Identity-3

Proof:   

So that-

Identity-4

Proof:

So that,

Identity-5                curl (u

Proof:

So that

Curl (u

Identity-6:

Proof:

So that-

Identity-7:

Proof:

So that-

Example-1: Show that-

1.

2.

Sol. We know that-

2. We know that-

= 0

Example-2: If then find the divergence and curl of .

Sol.  we know that-

Now-

Example-3: Prove that

Note- here is a constant vector and

Sol. Here    and

So that

Now-

So that-

Key takeaways-

1. 
2. 
3. 
4. Gradient of a constant (

If f and g are two scalar point functions, then

5.      If f and g are two scalar point functions, then

6.      If f and g are two scalar point functions, then-

7.     

8.     

9.      If then the vector is said to be irrotational.

10.  grad uv = u grad v + v grad u

11. 

12.  curl (u

6.4      Directional Derivatives, Angle between surfaces

Key takeaways-

1. The directional derivative of ϕ in the direction l, m, n= l + m+
2. The directional derivative of ϕ in the direction of =

6.5      Solenoidal and irrotational field

Solenoidal field-

A solenoidal field F is characterised by the following conditions-

1.

2. Flux along every closed surface is zero.

3.

Note- In an solenoidal field for which , the vector F can always be expressed as the curl of a vector function V.

So that-

Irrotational field-

An irrotatonal field F is characterised by the following conditions-

1.

2. Circulation along every closed surface is zero.

3.

Note- In an irrotational field for which , the vector F can always be expressed as the gradient of a scalar function provided the domain is simply connected.

So that-

Here the scalar function is called the potential.

Example-1: Prove that the vector field is irrotational and find its scalar potential.

Sol. As we know that if then field is irrotational.

So that-

So that the field is irrotational and the vector F can be expressed as the gradient of a scalar potential,

That means-

Now-

   ………………… (1)

  ……………………. (2)

Integrating (1) with respect to x, keep ‘y’ as constant-

We get-

      …………….. (3)

Integrating (1) with respect to y, keep ‘x’ as constant-

We get-

      …………….. (4)

Equating (3) and (4)-

and

So that-

Example-2: Prove that the vector field is solenoidal and irrotational.

Sol. We know that if  then the vector field will be solenoidal.

So that-

=

So that the vector field is solenoidal.

Now for irrotational field we need prove- 

So that-

Thus, the vector field F is irrotational.

Example-3: Show that the vector field is irrotational and find the scalar potential function.

Sol. Now for irrotational field we need prove- 

So that-

So that the vector field is irrotational.

Now in order to find the scalar potential function-

Key takeaways-

1. In an solenoidal field for which , the vector F can always be expressed as the curl of a vector function V.

So that-

2. An irrotatonal field F is characterised by the following conditions-

1.

2. Circulation along every closed surface is zero.

3.

3. In an irrotational field for which , the vector F can always be expressed as the gradient of a scalar function provided the domain is simply connected.

So that-

Here the scalar function is called the potential.

References

1. Erwin Kreyszig, Advanced Engineering Mathematics, 9thEdition, John Wiley & Sons, 2006.

2. N.P. Bali and Manish Goyal, A textbook of Engineering Mathematics, Laxmi Publications.

3. Higher engineering mathematic, Dr. B.S. Grewal, Khanna publishers

4. HK dass, engineering mathematics.