1211a4 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 1st Semester: Assignment 4 Due date: 2010 November 5 (Friday), 5:00 pm. 1. Calculate the flow line x ( t ) of the vector field F ( x, y, z ) = 2 y i 3 y j + z 3 k that passes through the point x (0) = (3 , 5 , 7) at t = 0. 2. Consider the vector field F = 2 x i 2 y j 3 k . (a) Show that F is a gradient field. (b) Describe the equipotential surfaces of F in words and with sketches. 3. If x is a flow line of a gradient vector field F = f , show that the function G ( t ) = f ( x ( t )) is an increasing function of t . 4. Prove Theorem 4.4 on Page 218 in the textbook. 5. Establish the following equalities: (a) ∇ × ( f F ) = f ∇ × F + f × F . (b) ∇ · ( F × G ) = G · ∇ × F F · ∇ × G . 6. The Laplacian operator, denoted by 2 , is the second-order partial differential operator defined by 2 = 2 ∂x 2 + 2 ∂y 2 + 2 ∂z 2 . (a) Explain why it makes sense to think of
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1211a4 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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