orick4 - F where F =(y cos(z i(x sin(z j(xy k and indicate...

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MATH 22 Exam 4 Name: ______________________ FALL, 2000 1. (8 pts.) Find the gradient vector field for the function f ( x , y , z ) = xy cos(ln( z )) 2. (14 pts.) Let C be the path consisting of parabolic arc along y = 2x 2 from (0, 0) to (1, 2). Calculate the line integrals: a. x ds C b. ydx + x dy C 3. (16 pts.) Evaluate the line integral F d r C where F = (yz + 2x) i + (xz + 3y 2 ) j + (xy) k , and r (t) = (t) i + (tln(t)) j + (e t ) k with 1 = t = 2. 4. (12 pts.) Use Green’s Theorem to evaluate the integral (1 + cot( x - π 2 )) dx + ( x 2 + e y ) dy C where C is the positively oriented boundary of the region enclosed by the curves y = 0, x = 1, and y = x . 5. (14 pts.) Find curl F , div F , and div(curl F ) where
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Unformatted text preview: F ) where F = (y cos(z)) i + (x sin(z)) j + (xy) k and indicate whether or not the vector field F is conservative. Justify your answer. 6. (12 pts.) Find and equation for the tangent plane to the parametric surface r (u,v) = v 2 i + u 2 j- uv k with 1 = u = 2, and -2 = v = 2sin(u p/2). At the point (1, 1, -1). 7. (12 pts.) Set up but do not evaluate an integral for the surface area of the parametric surface r (u,v) = u 2 i- uv j + v 2 k with 1 = u = 2, and -2 = v = 2sin(u p/2). 8. (12 pts.) Evaluate the surface integral xydS S ∫∫ where S is the part of the plane z = 2y + x inside the cylinder x 2 + y 2 = 4...
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