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Lecture Notes From Math 32B, Lecture notes of Mathematics

University of California - Los Angeles (UCLA)Mathematics

Lecture Notes From Math 32B. Very detailed and very useful

Typology: Lecture notes

2024/2025

Uploaded on 06/06/2025

hoi-man-yeung
hoi-man-yeung 🇺🇸

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Math 32B Discussion 5
1. Let Dbe the region shown below.
(a) Consider the change of variables (u, v) = F(x, y) = (xy, y/x). Sketch the region
D0=F(D) in the (u, v) plane.
(b) Find the inverse map G(u, v) = (x, y), and compute Jac(G).
(c) Using the change of variables formula, compute RRDy2dA by doing a double inte-
gral over D0.
pf2

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Math 32B 1. Let D be the region shown below. Discussion 5

(a) Consider the change of variables ( D 0 = F (D) in the (u, v) plane. u, v) = F (x, y) = (xy, y/x). Sketch the region (b) Find the inverse map G(u, v) = (x, y), and compute Jac(G). (c) Using the change of variables formula, computegral over D 0. ∫∫ D y^2 dA by doing a double inte-

Math 32B 2. Let using the change of variables D be the region inside the ellipse uDiscussion 5 = x + xy, v 2 + 2 =xy y −+ 2.y 2 − 4 y = 8. Compute the area of D

  1. Determine whether each of the following vector fields is divergence-free and curl-free.(a) F(x, y, z) = 〈x − yz, −xz + yz (^2) , −xy + y (^2) z〉 (b) F(x, y, z) = 〈x sin(xz), y cos(yz), −z cos(yz) − z sin(xz)〉 (c) F(x, y, z) = 〈x/r^3 , y/r^3 , z/r^3 〉, where r = √x^2 + y^2 + z^2.