Calculus - Standalone book
3rd Edition
ISBN: 9781464125263
Author: Jon Rogawski, Colin Adams
Publisher: W.H. Freeman & Co
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Chapter 18.1, Problem 3PQ
To determine
To show: if the circulation of a conservative
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(12 points) Let
E={(x, y, z)|x²+ y² + z² ≤ 4, x, y, z > 0}.
(a) (4 points) Describe the region E using spherical coordinates, that is, find p, 0, and such
that
(x, y, z) (psin cos 0, psin sin 0, p cos) € E.
(b) (8 points) Calculate the integral
E
xyz dV using spherical coordinates.
(10 points) Let f(x, y, z) = ze²²+y². Let
E = {(x, y, z) | x² + y² ≤ 4,2 ≤ z < 3}.
Calculate the integral
y,
f(x, y, z) dV.
(14 points) Let f: R3 R and T: R3.
→R³ be defined by
f(x, y, z) = ln(x²+ y²+2²),
T(p, 0,4)=(psin cos 0, psin sin, pcos).
(a) (4 points) Write out the composition g(p, 0, 4) = (foT)(p,, ) explicitly. Then calculate the
gradient Vg directly, i.e. without using the chain rule.
(b) (4 points) Calculate the gradient Vf(x, y, z) where (x, y, z) = T(p, 0,4).
(c) (6 points) Calculate the derivative matrix DT(p, 0, p). Then use the Chain Rule to calculate
Vg(r,0,4).
Chapter 18 Solutions
Calculus - Standalone book
Ch. 18.1 - Prob. 1PQCh. 18.1 - Prob. 2PQCh. 18.1 - Prob. 3PQCh. 18.1 - Prob. 4PQCh. 18.1 - Prob. 1ECh. 18.1 - Prob. 2ECh. 18.1 - Prob. 3ECh. 18.1 - Prob. 4ECh. 18.1 - Prob. 5ECh. 18.1 - Prob. 6E
Ch. 18.1 - Prob. 7ECh. 18.1 - Prob. 8ECh. 18.1 - Prob. 9ECh. 18.1 - Prob. 10ECh. 18.1 - Prob. 11ECh. 18.1 - Prob. 12ECh. 18.1 - Prob. 13ECh. 18.1 - Prob. 14ECh. 18.1 - Prob. 15ECh. 18.1 - Prob. 16ECh. 18.1 - Prob. 17ECh. 18.1 - Prob. 18ECh. 18.1 - Prob. 19ECh. 18.1 - Prob. 20ECh. 18.1 - Prob. 21ECh. 18.1 - Prob. 22ECh. 18.1 - Prob. 23ECh. 18.1 - Prob. 24ECh. 18.1 - Prob. 25ECh. 18.1 - Prob. 26ECh. 18.1 - Prob. 27ECh. 18.1 - Prob. 28ECh. 18.1 - Prob. 29ECh. 18.1 - Prob. 30ECh. 18.1 - Prob. 31ECh. 18.1 - Prob. 32ECh. 18.1 - Prob. 33ECh. 18.1 - Prob. 34ECh. 18.1 - Prob. 35ECh. 18.1 - Prob. 36ECh. 18.1 - Prob. 37ECh. 18.1 - Prob. 38ECh. 18.1 - Prob. 39ECh. 18.1 - Prob. 40ECh. 18.1 - Prob. 41ECh. 18.1 - Prob. 42ECh. 18.1 - Prob. 43ECh. 18.1 - Prob. 44ECh. 18.1 - Prob. 45ECh. 18.1 - Prob. 46ECh. 18.1 - Prob. 47ECh. 18.2 - Prob. 1PQCh. 18.2 - Prob. 2PQCh. 18.2 - Prob. 3PQCh. 18.2 - Prob. 4PQCh. 18.2 - Prob. 5PQCh. 18.2 - Prob. 1ECh. 18.2 - Prob. 2ECh. 18.2 - Prob. 3ECh. 18.2 - Prob. 4ECh. 18.2 - Prob. 5ECh. 18.2 - Prob. 6ECh. 18.2 - Prob. 7ECh. 18.2 - Prob. 8ECh. 18.2 - Prob. 9ECh. 18.2 - Prob. 10ECh. 18.2 - Prob. 11ECh. 18.2 - Prob. 12ECh. 18.2 - Prob. 13ECh. 18.2 - Prob. 14ECh. 18.2 - Prob. 15ECh. 18.2 - Prob. 16ECh. 18.2 - Prob. 17ECh. 18.2 - Prob. 18ECh. 18.2 - Prob. 19ECh. 18.2 - Prob. 20ECh. 18.2 - Prob. 21ECh. 18.2 - Prob. 22ECh. 18.2 - Prob. 23ECh. 18.2 - Prob. 24ECh. 18.2 - Prob. 25ECh. 18.2 - Prob. 26ECh. 18.2 - Prob. 27ECh. 18.2 - Prob. 28ECh. 18.2 - Prob. 29ECh. 18.2 - Prob. 30ECh. 18.2 - Prob. 31ECh. 18.2 - Prob. 32ECh. 18.2 - Prob. 33ECh. 18.2 - Prob. 34ECh. 18.2 - Prob. 35ECh. 18.2 - Prob. 36ECh. 18.3 - Prob. 1PQCh. 18.3 - Prob. 2PQCh. 18.3 - Prob. 3PQCh. 18.3 - Prob. 4PQCh. 18.3 - Prob. 5PQCh. 18.3 - Prob. 1ECh. 18.3 - Prob. 2ECh. 18.3 - Prob. 3ECh. 18.3 - Prob. 4ECh. 18.3 - Prob. 5ECh. 18.3 - Prob. 6ECh. 18.3 - Prob. 7ECh. 18.3 - Prob. 8ECh. 18.3 - Prob. 9ECh. 18.3 - Prob. 10ECh. 18.3 - Prob. 11ECh. 18.3 - Prob. 12ECh. 18.3 - Prob. 13ECh. 18.3 - Prob. 14ECh. 18.3 - Prob. 15ECh. 18.3 - Prob. 16ECh. 18.3 - Prob. 17ECh. 18.3 - Prob. 18ECh. 18.3 - Prob. 19ECh. 18.3 - Prob. 20ECh. 18.3 - Prob. 21ECh. 18.3 - Prob. 22ECh. 18.3 - Prob. 23ECh. 18.3 - Prob. 24ECh. 18.3 - Prob. 25ECh. 18.3 - Prob. 26ECh. 18.3 - Prob. 27ECh. 18.3 - Prob. 28ECh. 18.3 - Prob. 29ECh. 18.3 - Prob. 30ECh. 18.3 - Prob. 31ECh. 18.3 - Prob. 32ECh. 18.3 - Prob. 33ECh. 18.3 - Prob. 34ECh. 18.3 - Prob. 35ECh. 18.3 - Prob. 36ECh. 18.3 - Prob. 37ECh. 18.3 - Prob. 38ECh. 18.3 - Prob. 39ECh. 18.3 - Prob. 40ECh. 18.3 - Prob. 41ECh. 18.3 - Prob. 42ECh. 18 - Prob. 1CRECh. 18 - Prob. 2CRECh. 18 - Prob. 3CRECh. 18 - Prob. 4CRECh. 18 - Prob. 5CRECh. 18 - Prob. 6CRECh. 18 - Prob. 7CRECh. 18 - Prob. 8CRECh. 18 - Prob. 9CRECh. 18 - Prob. 10CRECh. 18 - Prob. 11CRECh. 18 - Prob. 12CRECh. 18 - Prob. 13CRECh. 18 - Prob. 14CRECh. 18 - Prob. 15CRECh. 18 - Prob. 16CRECh. 18 - Prob. 17CRECh. 18 - Prob. 18CRECh. 18 - Prob. 19CRECh. 18 - Prob. 20CRECh. 18 - Prob. 21CRECh. 18 - Prob. 22CRECh. 18 - Prob. 23CRECh. 18 - Prob. 24CRECh. 18 - Prob. 25CRECh. 18 - Prob. 26CRECh. 18 - Prob. 27CRECh. 18 - Prob. 28CRECh. 18 - Prob. 29CRECh. 18 - Prob. 30CRECh. 18 - Prob. 31CRECh. 18 - Prob. 32CRECh. 18 - Prob. 33CRECh. 18 - Prob. 34CRECh. 18 - Prob. 35CRECh. 18 - Prob. 36CRECh. 18 - Prob. 37CRECh. 18 - Prob. 38CRECh. 18 - Prob. 39CRECh. 18 - Prob. 40CRECh. 18 - Prob. 41CRE
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