1. What are the contours of f(x,y) = x-3y?
Straight lines of gradient -3
Straight lines of gradient 3
Straight lines of gradient -1/3
Straight lines of gradient 1/3

2. Which of the following contour plots best represents φ(x,y) = 2x + y ?1. 1 2 3 2.
None of the above 3.

3. If C is a closed contour, what symbol is used when evaluating a line integral?1 2 3 4 1. 2. 3. 4.

4. Which of the following diagrams represents the unit vectors i and j?1 2 3 4 1. 2. 3. 4.

5. φ(x,y,z)= xy - x²yz² + y²z Find gradφ
xyi - x²yz²j + y²zk
yi - x²z²j – 2x²y²zk
(y – 2yz²)i + (x - x²z² + 2z)j + y²k
(y – 2xyz²)i + (x - x²z² + 2yz)j + (y²-2x²yz)k

6. φ = 3x + 2xy²z - y²z² Find gradφ
(3+2y²z)i + (2x–z)2yzj +(x–z)2y²k
(3+2y²z)xi + (2x–z)2y²zj +(x–z)2y²zk
3i + 4xyzj – 2y²zk
3i + 4xyzj + 2y²zk

7. gradf can also be written as which of the following?1 2 3 4 1. 2. 3. 4.

8. Which of the following shows positive divergence?1. 1 2 3 2. 3.

9. F = xi +xyj – xyzk Find divF
1 + x – xy
i + xj – xyk
x + y – xy
x + xy – xyz

10. v = axyi – ½y²j – yzk Find div v
(a + 2)y
(a – 2)y
(2 – a)y
a – 2y

11. Using div v = (a – 2)y, find a to ensure fluid is incompressible1 2 3

12. F = xi + xyj – xyzk Find curlF
(-z-y)xyi + (yz+1)xj +(y-1)xk
(-z-y)xyi - (yz+1)xj +(y-1)xk
-xzi – yzj + yk
- xzi + yzj + yk

13. F = 3yzi – xzj + 2xy²zk Find curlF
(4yz+1)xi + (2yz-3)yj - 4zk
(4yz+1)xi - (2yz-3)yj - 4zk
-2zi + (4yz+1)xj + (3-2yz)yk
-2zi – (4yz+1)xj + (3-2yz)yk

14. The curl of a vector field measures which of the following?
The greatest rate of change at any point
The rate at which fluid is flowing away from a point
The rotation at a point

15. Which of the following statements are true?
div(grad f) exists and is a scalar
grad(div v) exists and is a scalar
div(curl v) exists and is a vector
curl(div v) exists and is a vector

16. A = xyzi + xy²j - x²zk and φ = 3xyz² Find ( )φ
3xyz²(yzi + xyj – 2x²k)
3xyz²(yz + xy – 2x²)
3xyz²(yzi + 2xyj – x²k)
3xyz²(yz + 2xy – x²)

17. φ = 2xy²z³ - x²z² +yz. Find z(z + 4xz² +12xy²) z²i + 4xz³j + 12xy²zk
z²(2y + x + 4xyz + 6xy²z)
-2z² + 4xz³ + 12xy² - 2x²

18. If φ = x²y³z³. Find the rate of change of φ at (1,3,1) in the direction of2 3 4 1. 2. 3. 4.

19. Find where C is the closed path shown on the diagram below
Given
Find where C is the closed path shown on the diagram below 2 3 4
Don’t Know

20. r = 3ti + (8t – 2t²)j. Find 7t
3ti + 4tj
11 – 4t
3i + (8 – 4t)j

21. A magnetic field B is given by. Find1 2 3 4 1. 2. 3. 4.

22. Given
What is divF ? 1. 1 2 3 2. 3.

23. Find 1. 1 2 3 4 2. 3. 4.