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 14.3 - Partial Derivatives - Exercises 14.3 - Page 809: 74

Answer

The Laplace's equation is satisfied.

Work Step by Step

We need to compute the Laplace's equation and prove that it equals to $0$. In order to find the partial derivative, we will differentiate with respect to $x$, by keeping $y$ and $z$ as a constant, and vice versa: $f_x=-6xz$; $f_y=-6yz $ and $f_z=6z^2-3x^2-3y^2$ Consider the Laplace's equation $\nabla^2 f =\dfrac{\partial^2 f}{\partial x^2}+\dfrac{\partial^2 f}{\partial y^2}+\dfrac{\partial^2 f}{\partial z^2}$ $\nabla^2 f =-6z-6z+12z=0$ Thus, the Laplace's equation is satisfied.
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