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HW9 - MA/PH 607 Howell Homework Handout IX A For the...

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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 " # $ % lparen1 rparen1 lparen1 rparen1 !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 BßC C œB each point in satisfies . lparen1 rparen1 V # # In addition, assume and are, respectively, a scalar field and a vector field with 9 F coordinate formulas 9 lparen1 rparen1 lparen1 rparen1 ˆ BßC œ B C BßC œ #BC mathplus BmathminusC # # 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 mathminus !ß! #ß% !ß! # $ # $ lparen1 rparen1 lparen1 rparen1 lparen1 rparen1 b. I V V œ mathminus " %
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Homework Handout IX page 2 B. For these problems, we are using cylindrical coordinates in three- e f lparen1 rparen1 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 lparen1 rparen1 lparen1 rparen1 lparen1 rparen1 lparen1 rparen1 > 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 lparen1 rparen1 lparen1 rparen1 lparen1 rparen1 lparen1 rparen1 lparen1 rparen1 > µ > ß > ß DÐ>Ñ œ $ß # >ß > !lessthan>lessthan& 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 lparen1 rparen1 <ß ßD œ Ð Ñ mathplus D mathplus 9 9 cos 3 9 3 . 3. r r Compute and when is the semicircle paramtrized by ' ' V V 9 e e < ‚. ‚. V r lparen1 rparen1 lparen1 rparen1 lparen1 rparen1 > µ ß ß D œ $ß ß & !lessthan lessthan 3 9 9 9 1 for . C. r Determine the formulas for and in terms of the standard spherical coordinate . .= system e f lparen1 rparen1 <ß ß Þ ) 9 D. r Determine the formulas for and in terms of the parabolic coordinate system . .= e f lparen1 rparen1 ?ß@ from problem in . K Homework Handout VI E. Arfken & Weber, page 59 : 1 (note: some positive constant), 2, 3 (These three 5 œ
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