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Midterm 1 Solution

Midterm 1 Solution - Name Math 362 Spring 2006 Professor...

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Unformatted text preview: Name: Math 362, Spring 2006, Professor Callahan Test #1, Thu–Fri, Feb. 23–24. Note 1: This test is closed book. You may use one 8 1 2 × 11 sheet of notes (both sides). Note 2: Show your work. Clarity counts. If I can’t follow your reasoning I can’t give credit. Problem 1: Evaluate contintegraltext C u · d r , where u = ( x 3 + 2 xy, x 2 y ) and C is the curve that goes from ( − 1 , 0) to (1 , 0) along the parabola y = 1 − x 2 . Answer: We parametrize the curve by r ( x ) = ( x, x 2 ) for − 1 ≤ x ≤ 1. Then d r = (1 , 2 x ) dx , so that u · d r = ( x 3 + 2 xy, x 2 y ) · (1 , 2 x ) dx = ( x 3 + 2 x 3 , x 4 ) · (1 , 2 x ) dx = (3 x 3 + 2 x 5 ) dx, and then integraldisplay C u · d r = integraldisplay 1- 1 (3 x 3 + 2 x 5 ) dx = parenleftbigg 3 x 4 4 + 2 x 6 6 parenrightbiggvextendsingle vextendsingle vextendsingle vextendsingle 1- 1 = 3 4 + 1 3 − 3 4 − 1 3 = 0 . 1 Problem 2: Write the following equations in Einstein subscript notation: ( a ): ∇ · a = b · ( c ×...
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Midterm 1 Solution - Name Math 362 Spring 2006 Professor...

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