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# takehome-s - 6(a Direct method 2 On the surface has...

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1 SOLUTIONS TO MATH 223 TAKE HOME TEST Spring 2007 1. The component of does not contribute to the line integral. Since the line integral of around is negative, for the line integral of to be positive, we need . No restriction on is needed. 2. For any value of , the vector ±eld is perpendicular to the vector which is in the direction of . Thus can take any value. The component of is in the direction of if the coef±cient of is positive, that is, if . 3. Since is perpendicular to and on , we have Area of surface 4. The plane is . Since the vector ±eld is on the plane, it is parallel to the normal and in the same direction, and of length 2 there. Thus Flux Area of square 5. (a) (i) By the de±nition of divergence, if is a sphere centered at , we have Vol Volume of Thus, if the sphere is small Voume of Since , if is the sphere given (ii) The relationship between ²ux, divergence, and volume holds when is cube. Thus Voume of gives (b) The point is a source because the ²ux out of a small region around the point is pos- itive; is a sink because the ²ux out of a small region around the point is negative.

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Unformatted text preview: 6. (a) Direct method: 2 On the surface, has no-component, so the-component of does not contribute to the Fux. Thus Since is perpendicular to and on , we have Area of (b) Using Stokes’ theorem, we replace the Fux integral by two line integrals around the circular boundaries, and , of . See ±igure 1. On , the left boundary, , so , and therefore . On , the right boundary, , so . This vector ²eld has and is tangent to the boundary and pointing in the same direction as . Thus Length of Figure 1 7. (a) I Green’s Theorem can be used. The curve is closed and the vector ²eld is smooth throughout the interior of the region enclosed II Green’s Theorem cannot be used. The vector ²eld is not de²ned at the origin which is inside the curve. III Green’s Theorem cannot be used. The curve is not closed. (b) ±or the integral in [I], let be the region enclosed by . See ±igure 2. Green’s Theorem gives 3 Figure 2...
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takehome-s - 6(a Direct method 2 On the surface has...

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