vcprac02 - , 4 , 5). Sketch the surface, and the normal you...

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The University of Sydney School of Mathematics and Statistics Practice Session 2 MATH2061: Vector Calculus Summer School 2012 1. (a) Find grad f if f ( x, y ) = x + 2 xy 3 y 2 . (b) Find grad g if g ( x, y, z ) = e x cos( yz 2 ). (c) Find φ if φ = 1 r x 2 + y 2 + z 2 . 2. (a) Given that F = 2 xy i + ( x 2 + z 2 ) j + (2 yz + 2 z ) k is conservative, ±nd a scalar function φ such that F = φ . (b) Find the work done by F along any path C from the point P (1 , 1 , 1) to the point Q (1 , 2 , 3) . 3. Let S be the surface de±ned by z = x 2 + y 2 . (a) Find the outward normal to this surface at the point P : (1 , 1 , 2). (b) Find the equation of the tangent plane to S at P . 4. Find a unit vector normal to the cone z = r x 2 + y 2 at the point (3
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Unformatted text preview: , 4 , 5). Sketch the surface, and the normal you have found. 5. Calculate Curl F if F = x 2 z i y j + z 3 k . 6. Let F be the vector eld F = ze x sin y i + ze x cos y j + e x sin y 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 3 from t = 0 to t = 1. (c) Evaluate i C F d r where C is the unit circle, centre (0 , 0), taken once anti-clockwise. Copyright c c 2012 The University of Sydney...
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