Tutorial 3

# Tutorial 3 - s and t Suppose further that when s,t =(2 1 ∂y/∂t = 0 Determine ∂z ∂t(2 1 4 If w = f ± x 2-y 2 x 2 y 2 ² is a

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MATH1211/10-11(3))/Tu3 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 First Semester: Tutorial 3 1. Verify the product and quotient rules for the pair of functions f ( x,y ) = e xy and g ( x,y ) = x sin 2 y . 2. Consider the function F ( x,y,z ) = x 3 y 2 - 2 xyz 4 + e xz . (a) Find F xx , F yy , and F zz . (b) Calculate the mixed second-order partials F xy , F yx , F xz , F zx , F yz , and F zy , and verify Theorem 4.3. (c) Is F xyx = F xxy ? Could you have known this without resorting to calculation? (d) Is F xyz = F yzx ? 3. Suppose that z = x 2 + y 3 , where x = st and y is a function of
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Unformatted text preview: s and t . Suppose further that when ( s,t ) = (2 , 1), ∂y/∂t = 0. Determine ∂z ∂t (2 , 1). 4. If w = f ± x 2-y 2 x 2 + y 2 ² is a diﬀerentiable function of u = x 2-y 2 x 2 + y 2 , show that then x ∂w ∂x + y ∂w ∂y = 0 . 5. Let f ( x,y ) = x 2-3 y 2 , g ( s,t ) = ( st, s + t 2 ). Calculate D ( f ◦ g ) in two ways: (a) by ﬁrst evaluating f ◦ g and (b) by using the chain rule and the derivative matrices Df and D g . 1...
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## This note was uploaded on 05/04/2011 for the course MATH 1211 taught by Professor Wang during the Spring '11 term at HKU.

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