What are the contours of f(x,y) = x-3y?

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Presentation transcript:

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

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

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

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

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

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

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

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

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

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

Using div v = (a – 2)y, find a to ensure fluid is incompressible 1 2 3

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

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

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

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

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)

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

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

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

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

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

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

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