undefined | unit 5 vector calculus

Back to Study material

Math

Unit-5Vector CalculusQuestion-1: If

and

then find-1.

Sol. 1. We know that-

Question-2: If

, then show that1.

Sol.

Suppose and

Now taking L.H.S,

Which is

Hence proved.

So that

Question-3: 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.Question-4: Show that-1.

Sol. We know that-

2. We know that-

= 0Question-5: Prove that

Note- here

is a constant vector and

Sol. here

and

So that

Now-

So that-

Question-6: What is the curl of the vector field F= ( x +y +z ,x-y-z,

)?Solution:Curl F =

= (2y+1)i-(2x-1)j+(1-1)k= (2y+1)i+(1-2x)j+0k= (2y+1, 1-2x,0)Question-7: Find the directional derivative of 1/r in the direction

where

Sol. Here

Now,

And

We know that-

So that-

Now,

Directional derivative =

Question-8: Find the directional derivatives of

at the point P(1, 1, 1) in the direction of the line

Sol. Here

Direction ratio of the line

are 2, -2, 1Now directions cosines of the line are-

Which are

Directional derivative in the direction of the line-

Question-9: 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-

Questin-10: 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-

Question-11: Evaluate

where

= (2xy +z2) I +x2j +3xz2 k along the curve x=t, y=t2, z= t3 from (0,0,0) to (1,1,1). Solution : F x dr =

Put x=t, y=t2, z= t3Dx=dt ,dy=2tdt, dz=3t2dt. F x dr =

=(3t4-6t8) dti – ( 6t5+3t8 -3t7) dt j +( 4t4+2t7-t2)dt k

=t4-6t3)dti –(6t5+3t8-3t7)dt j+(4t4 + 2t7 – t2)dt k

Question-12: Evaluate

if V is the region in the first octant bounded by

and the plane x = 2 and

.Sol.

x varies from 0 to 2The volume will be-

Question-13: Verify green’s theorem in xy-plane for

where C is the boundary of the region enclosed by

Sol. On comparing with green’s theorem, We get- P =

and Q =

and

By using Green’s theorem-

………….. (1)

And left hand side=

………….. (2)Now,Along

Along

Put these values in (2), we get- L.H.S. = 1 – 1 = 0So that the Green’s theorem is verified.Question-14: Show that

SolBy divergence theorem,

..…(1)Comparing this with the given problem let

Hence, by (1)

…………. (2)

Now ,

Hence, from (2), We get,

Question-15: Verify stoke’s theorem when

and surface S is the part of sphere

, above the xy-plane.Sol.We know that by stoke’s theorem,

Here C is the unit circle-

So that-

Now again on the unit circle C, z = 0dz = 0Suppose,

And

Now

……………… (1)Now- Curl

Using spherical polar coordinates-

………………… (2)From equation (1) and (2), stoke’s theorem is verified.Question-16: If

and C is the boundary of the triangle with vertices at (0, 0, 0), (1, 0, 0) and (1, 1, 0), then evaluate

by using Stoke’s theorem.Sol. here we see that z-coordinates of each vertex of the triangle is zero, so that the triangle lies in the xy-plane and

Now, Curl

Curl

The equation of the line OB is y = xNow by stoke’s theorem,

Sign Up