HW8 - MA/PH 607 Howell 10/31/2011 Homework Handout VIII A....

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Unformatted text preview: MA/PH 607 Howell 10/31/2011 Homework Handout VIII A. In a previous problem, you sketched the curves with x l r l r l r > > > > > 3 9 # > 1 with x l r l r > > > > m > 3 9 # 1 $ with 4 x l r l r l r > > > % > 3 9 1 > Now: 1. Find in terms of the polar coordinates and the associated tangent vectors and . . .> x e e 3 9 Also, on your sketch of each curve, sketch . .> x at different points on the curve. 2. Find , the rate at which the arclength along the curve varies as varies .= .> > use your formula for . Then write out the integral that gives the length of the . .> B curve traced out at goes from to (if the integral is simple enough, evaluate it!). > ! " 3. Find in terms of the polar coordinates and the associated tangent vectors and . .> # # x e 3 e 9 . (Doing this requires that youve found the acceleration formula for polar coordinates, which is exercise 9.3 in the online notes.) B. Let be the first coordinate system in the e f l r ? @ Three Coordinate Systems for the Plane handout, and let be the position of George the Gerbil at time . x > ?> @> l r > 1. Find the general formulas for Georges velocity, speed and acceleration, . .= . .> .> .> x x , and # # . 2. Assume and do the following: l r l r ?> @> #> $> a. Sketch the resulting curve on the plane. b. Compute Georges velocity, speed and acceleration at each time . > c. Set up and compute (if practical) the integral giving the distance traveled by George between and . > ! > $ 3. Assume and do the following: l r l r ?> @> > > # a. Sketch the resulting curve on the plane. b. Compute Georges velocity, speed and acceleration at each time . > c. Set up and compute (if practical) the integral giving the distance traveled by George between and . > ! > $ Homework Handout VIII page 2 C. K Let be the parabolic coordinate system for the plane from exercise of Homework e f l r ? @ Handout VII . Using your results from that exercise, find the Christoffel symbols for this coordinate system and the partial derivatives ` ` ` ` `? `@ `? `@ , , and F F F F & & & & ? ? @ @ in terms of this coordinate system and . e f F F & & ? @ D. B Redo exercise , above, but using the parabolic coordinate system for the plane e f l r ? @ from exercise of Homework . Make use of the results from the last problem, K Handout VII above. E. M Let be the coordinate system for the plane from exercise of Homework e f l r ? @ Handout VII with . Using your results from that exercise, find the Christoffel symbols for this + " coordinate system and the partial derivatives ` ` ` ` `? `@ `? `@ , , and F F F F & & & & ? ? @ @ in terms of this coordinate system and . e f F F & & ? @ F. B Redo exercise , above, but using the coordinate system for the plane from e f l r ? @ exercise of Homework . Make use of the results from the last problem,...
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HW8 - MA/PH 607 Howell 10/31/2011 Homework Handout VIII A....

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