Hw4_sol - Mathematics 20E Spring 2009 Homework 4 Please complete the following problems and turn them in to the homework drop box in the sixth

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Unformatted text preview: Mathematics 20E - Spring 2009 - Homework 4 May 26, 2009 Please complete the following problems and turn them in to the homework drop box in the sixth floor of AP&M labelled “Math 20E / Ryan Szypowski” by Friday, May 22 at 5:30pm. Only a subset of the problems will be graded; leave some blank at your own risk. 1. Let F : Ê 3 → Ê 3 be given by F ( x, y, z ) = parenleftbigg ye z , xe z + 1 z , xye z − y z 2 parenrightbigg and let c : bracketleftbig , π 2 bracketrightbig → Ê 3 be a parameterization of an oriented curve C , given by c ( t ) = ( cos t sin t, sin 2 t, cos t + 1 ) . Use some trick to compute integraltext C F · d s . Solution : This problem is trying to have us realise that the given F is the gradient of some vector field and hence is path independant. To this effect, notice that ∇ × F = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle i jk ∂ ∂x ∂ ∂y ∂ ∂z ye z xe z + 1 z xye z − y z 2 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = . In fact, F = ∇ f where f ( x, y, z ) = xye z + y z . 1 Thus, we can compute integraldisplay C F · d s = f parenleftBig c parenleftBig π 2 parenrightBigparenrightBig − f ( c (0)) = f (0 , 1 , 1) − f (0 , , 2) = 1 . 2. Let v : Ê 3 → Ê 3 be given by v ( x, y, z ) = y i + z j + x k and let T : [0 , 1] × [0 , 1] → Ê 3 be a parameterization of an oriented surface S , given by T ( u, v ) = u i + uv j + ( u − v ) k ....
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This note was uploaded on 10/22/2009 for the course CHEM 140A taught by Professor Whiteshell during the Spring '04 term at UCSD.

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Hw4_sol - Mathematics 20E Spring 2009 Homework 4 Please complete the following problems and turn them in to the homework drop box in the sixth

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