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.
 Spring '10
 phfiier

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