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Question 6 Let O, A and B be the points (0, 0, 1), (1, 0, 1) and (0, 1, 1) respectively in the (x, y, z) plane. Let C be the closed curve OABO consisting of the line segment linking points O and A, followed by the quarter-circle x² + y² = 1 with centre the point O lying on the plane z = 1 and linking points A and B, followed by the line segment linking points B and O. Let F = (z ? 2y)i ? 2xj + xk. a) Calculate ? × F and comment on your result. [5 Marks] b) Calculate the line integral ?_C F · dl (directly) where the curve C is transversed in the counter-clockwise direction as seen from above. Hint: You may find the following trigonometric identity useful: cos 2? = cos² ? ? sin² ?. [10 Marks] c) By constructing a suitable potential function show that F is conservative. Hence confirm your result in part (b) above. [5 Marks]

          Question 6

Let O, A and B be the points (0, 0, 1), (1, 0, 1) and (0, 1, 1) respectively in the (x, y, z) plane. Let C be the closed curve OABO consisting of the line segment linking points O and A, followed by the quarter-circle x² + y² = 1 with centre the point O lying on the plane z = 1 and linking points A and B, followed by the line segment linking points B and O. Let F = (z ? 2y)i ? 2xj + xk.

a) Calculate ? × F and comment on your result. [5 Marks]

b) Calculate the line integral ?_C F · dl (directly) where the curve C is transversed in the counter-clockwise direction as seen from above. Hint: You may find the following trigonometric identity useful: cos 2? = cos² ? ? sin² ?. [10 Marks]

c) By constructing a suitable potential function show that F is conservative. Hence confirm your result in part (b) above. [5 Marks]

Question 6

Let O, A and B be the points (0, 0, 1), (1, 0, 1) and (0, 1, 1) respectively in the (x, y, z) plane. Let C be the closed curve OABO consisting of the line segment linking points O and A, followed by the quarter-circle x² + y² = 1 with centre the point O lying on the plane z = 1 and linking points A and B, followed by the line segment linking points B and O. Let F = (z ? 2y)i ? 2xj + xk.

a) Calculate ? × F and comment on your result. [5 Marks]

b) Calculate the line integral ?C F · dl (directly) where the curve C is transversed in the counter-clockwise direction as seen from above. Hint: You may find the following trigonometric identity useful: cos 2? = cos² ? ? sin² ?. [10 Marks]

c) By constructing a suitable potential function show that F is conservative. Hence confirm your result in part (b) above. [5 Marks]

Added by Brenda G.

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