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1 ProbSheet 1

This document contains a problem sheet with 17 multi-part questions regarding vector calculus and vector fields. The questions involve finding derivatives of scalar and vector functions, directional derivatives, divergence, curl, and laplacians. An answer key is provided with solutions to selected questions.

Uploaded by

Fatmah El Wardagy
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© Attribution Non-Commercial (BY-NC)
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Download as PDF, TXT or read online on Scribd
65 views2 pages

1 ProbSheet 1

This document contains a problem sheet with 17 multi-part questions regarding vector calculus and vector fields. The questions involve finding derivatives of scalar and vector functions, directional derivatives, divergence, curl, and laplacians. An answer key is provided with solutions to selected questions.

Uploaded by

Fatmah El Wardagy
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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HG2M13

________________________________________________________________________

Problem Sheet 1

Additional Questions

1.

If = 2 xz 4 x 2 y , find and || at the point (2, -2, -1).

2.

If A = 2 x 2 i 3 yzj + xz 2 k and = 2z x3y, find A and A at the


point (1, -1, 1).

3.

If F = x 2 z + e y

and G = 2 z 2 y xy 2 , find

i.
(F - G)
at the point (1, 0, -2).

ii.

(FG)

4.

Find |r|3

5.

Prove that
i.
(F + G) = F + G
ii.
(FG) = F G + G F
where F and G are differentiable scalar functions of x, y and z.

6.

Find the directional derivative of = x 2 yz + 4 xz 2 at (1, -2, -1) in the


direction of 2i j 2k.

7.

Find the direction vector where f ( x, y , z ) = xyz 3 y 2 increases most rapidly


at the point (1, 1, 3) and also the maximum rate of change of f .

8.

Find a vector that is normal to the level curve xy 2 3 x 2 y = 10 at the point

(1, 5) .
9.

Determine the unit normal vector to the surface xz 2 3 xyz 7 x = 2 at the


point (1, 0, 3) .

10.

If A = 3 xyz 2 i + 2 xy 3 j x 2 yz k and = 3 x 2 yz , find


i. A
iv. ()
at the point (1, -1, 1).

ii. A
v. A

iii. (A)

11.

If = 3 x 2 z y 2 z 3 + 4 x 3 y + 2 x 3 y 5 , find 2 .

12.

Verify

that

the

vector

field

F( x, y, z ) = ( y sin z ) i + xz j + xe y k

incompressible.

_____________________________________________________________________ 1

is

HG2M13
________________________________________________________________________
13.

Is the vector field F( x, y , z ) = ( ze x + sin y )i + ( x cos y ) j + e x k irrotational?


Explain.

14.

Verify that the vector field F( x, y , z ) = (e y + yz ) i + ( xe y + xz ) j + ( xy 3) k is


conservative and find its potential function.

15.

Find

the

values

of

a and b such that the


F ( x, y, z ) = (4 xy + ayz ) i + (2 x 2 + 3 xz ) j + bxy k is irrotational.

vector

field

16.

Let A = A ( x, y, z ), B = B( x, y, z ) and = ( x, y, z ) . Assuming all derivatives


involved exist and are continuous, prove that
i.
(A + B) = A + B
ii.
(A) = () A + ( A)

17.

If A = 2 xz 2 i yzj + 3 xz 3 k and = x 2 yz , find


ii. curl (A)
i. A
v. curl grad (A)
at the point (1, 1, 1).

iii. ( A)

iv. [A curl A]

Answers
1. 10i 4j 16k, 2 93

2. 5, 7i j 11k

3. i. -4i + 9j + k

ii. -8j

4. 3rr

6. 37/3

8.

9.

5i + 7 j

10. i. 4

n =

ii. -15

11.

6 z + 24 xy 2 z 3 6 y 2 z

15.

a = 3, b = 3

17.

i. i + j

7.

3i 3 j + k ;

19

1
(2i 9 j + 6k )
11

iii. 1

iv. 6
14.

v.

i 8 j 5k

f ( x, y, z ) = xe y + xyz 3 z

ii. 5i 3j 4k

iii. 5i + 3k

iv.

-2i + j + 8k

v. 0

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