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Module MA1132 (Frolov), Advanced Calculus Tutorial Sheet 5 To be solved during the tutorial session Thursday/Friday, 18/19 February 2016 1. Use appropriate forms of the chain rule to find z = sin x cos 2y ; 2 ∂z ∂u where x = 2u + 3v , y = u3 − 2v 2 . 2. A function f (x1 , . . . , xn ) is said to be homogeneous of degree k if f (tx1 , . . . txn ) = tk f (x1 , . . . , xn ) for t > 0. Show that it satisfies n X ∂f xi = kf . ∂xi i=1 3. Let r = Pn i=1 xi ei and r = |r|, where ei form an orthonormal basis of vectors in Rn . Find ∂r , ∂xi i = 1, 2, · · · , n , and ∇r , |∇r| . 4. Let ur be a unit vector whose counterclockwise angle from the positive x-axis is θ, and let uθ be a unit vector 90% counterclockwise from ur . Show that if z = f (x, y), x = r cos θ, y = r sin θ, then 1 ∂z ∂z ur + uθ . ∇z = ∂r r ∂θ 5. Consider the surface p 3 2x2 − 3xy 2 + 3 cos(2x + 3y) − 3y 3 + 18 z = f (x, y) = ln 2 (a) Find an equation for the tangent plane to the surface at the point P (3, −2, z0 ) where z0 = f (3, −2). (b) Sketch the tangent plane. (c) Find parametric equations for the normal line to the surface at the point P (3, −2, z0 ). (d) Sketch the normal line to the surface at the point P (3, −2, z0 ). Show the details of your work. 1