M427Lmt - θ and y = r sin θ Show that ± ∂u ∂x 2 ± ∂u ∂y 2 = ± ∂u ∂r 2 1 r 2 ± ∂u ∂θ 2(This is a chain rule problem 8

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M427L Midterm Exam Name NO NOTES. NO CALCULATORS. 1. Find and classify the critical points of f ( x,y,z ) = x 2 + y 2 + z 2 + yz + xz. 2. Find the length of the curve c ( t ) = ( t cos t,t sin t, 2 2 3 t 3 / 2 ) , 0 t π. 3. Let F ( x,y,z ) = ( x 2 - y, 4 z,x 2 ). Find div F and curl F . 4. Express the equation x 2 + y 2 = 1 in spherical coordinates. 5. Find the point where the lines c 1 ( t ) = (1 - t, 2 t, - 3+2 t ) and c 2 ( t ) = (2 t +2 , 3 t +19 , 4 t +19) intersect. Then find the equation of the plain containing both lines. 6. Let f ( x,y ) = xy + 1. Write the derivative and Hessian of f at the point (2 , 4). Then use a quadratic approximation to estimate f (2 . 03 , 3 . 99). 7. Suppose u = f ( x,y ) and x = r cos
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Unformatted text preview: θ and y = r sin θ. Show that ± ∂u ∂x ¶ 2 + ± ∂u ∂y ¶ 2 = ± ∂u ∂r ¶ 2 + 1 r 2 ± ∂u ∂θ ¶ 2 . (This is a chain rule problem.) 8. Let f ( x,y ) = 3 x 2-y 2 . Find all unit vectors v such that D v f (1 , 2) = 0 . 9. Let F : R 3 → R 3 be a C 2 vector field. Show that ∇ • ( ∇ × F ) = 0 . 10. Find T , N , B ,κ, and τ for the curve c ( t ) = (3sin t, 4 t, 3cos t ) at the point where t = π/ 2. Sketch the curve and show these quantities on your sketch....
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This note was uploaded on 05/02/2011 for the course M 427L taught by Professor Keel during the Spring '07 term at University of Texas at Austin.

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