HW9 - MA/PH 607 Howell 11/7/2011 Homework Handout IX A. For...

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MA/PH 607 Howell 11/7/2011 Homework Handout IX A. For the following problems, we will use Cartesian coordinates to describe points in the plane, and will let , , and be the four curves from V V V V " # $ % l r l r !ß ! #ß % to indicated in the figure. Note that: \ V V " % and consist of portions of the and ] –axes, V $ is a straight line, and Bß C C œ B each point in satisfies . l r V # # In addition, assume and are, respectively, a scalar field and a vector field with 9 F coordinate formulas 9 l r l r ˆ Bß C œ B C Bß C œ #BC m B M C # # and . F i j 1. Find parametrizations for each of these four curves (the parametrizations for and V V " % can each be in “two pieces”). 2. r Determine for each of the parametrizations just found, and then compute . ( V 5 F † . 5 œ "ß #ß $ß % r for and . 3. Determine for each of the parametrizations just found, and then .= set up (computing if practical) ( V 5 9 .= 5 œ "ß #ß $ß % for and . 4. Using your results from the above, compute ( I F † . r where a. (I.e, take from to , then go backwards on to .) I V V V V œ M !ß ! #ß % !ß ! # $ # $ l r l r l r b. I V V œ M " %
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Homework Handout IX page 2 B. For these problems, we are using cylindrical coordinates in three- e f l r 3 9 ß ß D dimensional Euclidean space. 1. Suppose we have a curve in the plane with a cylindrical coordinate parametrization V ß r l r l r l r l r > µ > ß > ß DÐ>Ñ 3 9 . Determine the formulas for and in terms of , , , , and . . .= r . . .D .> .> .> 3 9 e e k 3 9 2. Assume that, in fact, our curve is given by V r l r l r l r l r l r > µ > ß > ß DÐ>Ñ œ $ ß # > ß > 3 9 1 for . a. Sketch the curve. b. Compute where the square of the distance from the origin to . ' V 9 9 .= Ð Ñ œ B B c. r Compute where is the vector field with cylindrical coordinate formula ' V F F † . F e e k l r <ß ß D œ Ð Ñ m D m 9 9 cos 3 9 3 . 3. r r Compute and when is the semicircle paramtrized by ' ' V V 9 e e < ‚ . ‚ . V r l r l r l r > µ ß ß D œ $ ß ! l l 3 9 9 9 1 for . C. r Determine the formulas for and in terms of the standard spherical coordinate . .= system e f l r <ß ß Þ ) 9 D. r Determine the formulas for and in terms of the parabolic coordinate system . .= e f l r ?ß @ from problem in . K Homework Handout VI E. Arfken & Weber, page 59 : 1 (note: some positive constant), 2, 3 (These three
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HW9 - MA/PH 607 Howell 11/7/2011 Homework Handout IX A. For...

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