Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 14: Partial Derivatives - Section 14.1 - Functions of Several Variables - Exercises 14.1 - Page 787: 12

Answer

$\{(x,y)\in \mathbb{R}^{2}\ \ |\ \ $ $x^{2}+y^{2}\lt 4,\ \ x^{2}+y^{2}\neq 3 \}$
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Work Step by Step

First, the argument of ln must be positive, $4-x^{2}-y^{2} \gt 0$. $4 \gt x^{2}+y^{2}$ is the inside of a circle about the origin, radius 2. (The circle itself is excluded, graphed with a dashed line.) Second, the denominator must not be zero, and since $\ln 1=0$ we exclude $(x,y)$ for which $4-x^{2}-y^{2}=1$ $3=x^{2}+y^{2}$ Any points on the circle with radius $\sqrt{3}$ are to be excluded (the circle is graphed with a dashed line). Domain: $\{(x,y)\in \mathbb{R}^{2}\ \ |\ \ $ $x^{2}+y^{2}\lt 4,\ \ x^{2}+y^{2}\neq 3 \}$
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