Consider the function where and Verify each of the following

Chapter 13, Problem 61

(choose chapter or problem)

Consider the function \(w=f(x, y)\), where \(x=r \cos \theta\) and \(y=r \sin \theta\). Verify each of the following.

(a) \(\begin{array}{l}

\frac{\partial w}{\partial x}=\frac{\partial w}{\partial r} \cos \theta-\frac{\partial w}{\partial \theta} \frac{\sin \theta}{r} \\

\frac{\partial w}{\partial y}=\frac{\partial w}{\partial r} \sin \theta+\frac{\partial w}{\partial \theta} \frac{\cos \theta}{r}

\end{array}\)

(b) \(\left(\frac{\partial w}{\partial x}\right)^{2}+\left(\frac{\partial w}{\partial y}\right)^{2}=\left(\frac{\partial w}{\partial r}\right)^{2}+\left(\frac{1}{r^{2}}\right)\left(\frac{\partial w}{\partial \theta}\right)^{2}\)

Text Transcription:

w=f(x,y)

x=rcostheta

y=rsintheta

fracpartialwpartialx=fracpartialwpartialrcostheta-fracpartialwpartialthetafracsinthetar\\fracpartialwpartialy=fracpartialwpartialrsintheta+fracpartialwpartialthetafraccosthetar(fracpartialwpartialx)^2+(fracpartialwpartialy)^2=(fracpartialwpartialr)^2+(frac1r^2)(fracpartialwpartialtheta)^2