vctut02 - C from the point P (0 , , 0) to the point Q (2 ,...

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The University of Sydney School of Mathematics and Statistics Tutorial 2 MATH2061: Vector Calculus Summer School 2012 1. Find grad f if f ( x, y ) = x 2 cos xy . 2. Find grad φ if φ ( x, y, z ) = 3 x + 4 y 8 z . 3. Find φ if φ ( x, y, z ) = e x y + 3 xyz . 4. Calculate Curl F if F = (sinh x ) i + (cosh y ) j xyz k . 5. Let S be the surface de±ned by z = xy . (a) Find a normal to this surface at the point P : (2 , 3 , 6). (b) Find the equation of the tangent plane to S at P . 6. Find a unit vector normal to the sphere x 2 + y 2 + z 2 = 4 at the point (1 , 1 , 2). Sketch the surface, and the normal you have found. 7. (a) Given that F = 3 x 2 i + 4 yz j + (2 y 2 5) k is conservative, ±nd a scalar function φ such that F = φ. (b) Find the work done by F along any path
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Unformatted text preview: C from the point P (0 , , 0) to the point Q (2 , 1 , 1) . 8. Let F be the vector eld F = (2 x + sin yz ) i + (2 y + xz cos yz ) j + (2 z + xy cos yz ) k . (a) Show that F is conservative and nd a potential function for F . (b) Evaluate i C F d r where C is the curve x = t , y = t 2 , z = t 4 from t = 0 to t = 1. Extra question for outside the tutorial 9. Determine whether or not the vector eld F is conservative. F = x r x 2 + y 2 + z 2 i + y r x 2 + y 2 + z 2 j + z r x 2 + y 2 + z 2 k . Copyright c c 2012 The University of Sydney...
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