Determine whether the given function satisfies a Laplace equation.
-$$f ( x , y , z ) = 5 x ^ { 2 } - 3 y ^ { 2 } - 2 z ^ { 2 }$$

Question

Quizplus · Accepted Answer

The Laplace equation in Cartesian coordinates is $\frac{\partial^2 f}{\partial x^2} +\frac{\partial^2 f}{\partial y^2} +\frac{\partial^2 f}{\partial z^2} = 0$. Taking second derivatives of $f(x,y,z)$ with respect to each variable, we get:

\begin{align*}
\frac{\partial^{2} f}{\partial x^{2}} &= 10 \
\frac{\partial^{2} f}{\partial y^{2}} &= -6 \
\frac{\partial^{2} f}{\partial z^{2}} &= -4
\end{align*}

Adding these up, we get:

$$\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial^{2} f}{\partial z^{2}}=10-6-4=0$$

Therefore, the function satisfies the Laplace equation.

Determine Whether the Given Function Satisfies a Laplace Equation f(x,y,z)=5x2−3y2−2z2f ( x , y , z ) = 5 x ^ { 2 } - 3 y ^ { 2 } - 2 z ^ { 2 }f(x,y,z)=5x2−3y2−2z2

Determine Whether the Given Function Satisfies a Laplace Equation $f ( x , y , z ) = 5 x ^ { 2 } - 3 y ^ { 2 } - 2 z ^ { 2 }$