F ( x , y , z ) = (x^2)y i + (1/3)(x^2) j + xy k and C is the curve of intersect

Question

F(x, y, z) = (x^2)yi + (1/3)(x^2)j + xyk

and C is the curve of intersection of the hyperbolic paraboloid z= y^2 - x^2 and the cylinder x^2 + y^2 = 1 oriented counterclockwise as viewed from above.

B. Graph both the hyperbolic paraboloid and the cylinder with domains chosen so that you can see the curve C and the surface.

C. Find parametric equations for C and use them to graph C. (Enter your answer as a comma-separated list of equations. Let x and y be in terms of t where t is between 0 and 2pi.

Explanation / Answer

Solution :

(a)   

     Note that curl F = <x, -y, 0>.
Moreover, let S be the surface inside C.

So, c F · dr
= s curl F · dS, by Stokes' Theorem
= <x, -y, 0> · <-(-2x), -(2y), 1> dA, using cartesian coordinates
= 2(x2 + y2) dA
= ( = 0 to 2) (r = 0 to 1) 2r2 * r dr d, via polar coordinates
= 2 (r = 0 to 1) 2r3 dr
= r4 {for r = 0 to 1}
=

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