317-w09-202-hw-04

317-w09-202-hw-04 - dy i with constants a,b,c,d ∈ R Find...

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Math 317, Section 202, Homework no. 4 (due Friday February 5, 2010) [7 problems, 28 points] Problem 1 (8 points) . Stewart, Chapter 17.1, Exercises 11-18 (1 point each). No explanation required; only the answer counts. Problem 2 (4 points) . Compute the field lines of the following vector fields and sketch them. (a) (2 points) F ( x,y ) = h 2 y, - x i , (b) (2 points) F ( x,y ) = h y, - sin x i . Problem 3 (2 points) . Compute the field lines of the vector field F ( x,y,z ) = h x,y, - z i . Problem 4 (4 points) . Stewart, Chapter 17.1, Exercises 29-32 (1 point each). No explanation required; only the answer counts. Problem 5 (2 points) . Consider the vector field F ( x,y ) = h ax + by,cx
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Unformatted text preview: + dy i with constants a,b,c,d ∈ R . Find a condition on these constants that holds if and only if F is of the form F = ∇ ϕ for some real valued function ϕ ( x,y ). [Hint: Think of Clairaut’s theorem.] Problem 6 (4 points) . Given a real valued function ϕ ( x,y,z ), show that its gradient ∇ ϕ is orthogonal to the level surfaces of ϕ . [Hint: You may use all results that were shown in class.] Problem 7 (4 points) . Sketch the level curves of the function ϕ ( x,y ) = x 2 + 1 4 y 2 and the field lines of its gradient ∇ ϕ in the same diagram....
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This note was uploaded on 03/18/2010 for the course MAATH 317 taught by Professor Phfiier during the Spring '10 term at UBC.

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