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Unformatted text preview: Math 223008 Exam # 3 – Take Home Part Due on Monday 4 April 2005 at the beginning of class. Partial credit is possible, but you must show all work. Name: I hereby testify that this is individual work. Signed: − → y−1 x − → − → 5. Consider the vector ﬁeld F = 2 i− 2 j 2 2 x + (y − 1) x + (y − 1) − → (a) Show that F is a conservative vector ﬁeld. − →→ r (b) Justify why Green’s theorem cannot be applied to ﬁnd C F · d− for the curve C described below (and traced in the counterclockwise direction). (Hint: Where is the singularity?) 2 2 1.5 1 0.5 1 0.5 0.5 1 −→ → r (c) Find the integral C F ·d− by constructing a reasonable curve that would simplify your work tremendously. Parametrize your “reasonable curve” then calculate the line integral. The alternative is to use the following parametrization of C : x(t) = (1 − sin t) cos t y (t) = 2 + (1 − sin t) sin t for 0 ≤ t ≤ 2π . 3 6. Convert to the appropriate coordinate system, then integrate completely.
1 0 0 √ 1 −z 2 √ 1−x2 −z 2 √ − 1−x2 −z 2 z dy dx dz 4 7. Change the order of integration to dy dx dz √ √
2 4 −y 2 4−x2 −y 2 f (x, y, z ) dz dx dy 0 0 0 5 − → − → − → 8. (a) Show that the vector ﬁeld F = (x3 − 3xy 2 ) i + (−3x2 y + y 3 ) j is conservative. (b) Find a potential function V (x, y, z ) corresponding to this vector ﬁeld. ...
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 Spring '10
 DICKSON
 Math

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