91Ó°ÊÓ

Exer. \(35-40:\) If \(u=f(x, y), v=g(x, y),\) and \(f\) and \(g\) are differentiable, prove the identity. \(\nabla(c u)=c \nabla u\) for a constant

Find the dimensions of the rectangular box of maximum volume with faces parallel to the coordinate planes that can be inscribed in the ellipsoid $$ 16 x^{2}+4 y^{2}+9 z^{2}=144 $$

A function \(f\) of \(x\) and \(y\) is harmonic if $$ \frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}=0 $$ throughout the domain of \(f\). Prove that the given function is harmonic. $$ f(x, y)=e^{-x} \cos y+e^{-y} \cos x $$

If \(f(x, y)=\frac{1}{4} x^{2}+\frac{1}{25} y^{2},\) sketch the level curve of \(f\) that contains \(P(0,5)\), and sketch \(\nabla f]_{P}\).

Graph, on the same coordinate plane, the level curves for \(k=1,2,\) and \(3,\) using polar coordinates. $$ f(x, y)=3 x^{4}+x^{2} y^{2}+y^{4} $$

If \(u=f(x, y), v=g(x, y),\) and \(f\) and \(g\) are differentiable, prove the identity. \(\nabla(u+v)=\nabla u+\nabla v\)

Refer to Exercise \(32 .\) Find the degree \(n\) of the homogeneous function \(f\) and verify the formula $$x f_{x}(x, y)+y f_{y}(x, y)=n f(x, y)$$ \(f(x, y)=x y e^{y / x}\)

Find \(h(x, y)=g(f(x, y))\) and use Theorem (16.7) to determine where \(h\) is continuous. $$ f(x, y)=x+\tan y ; \quad g(z)=z^{2}+1 $$

If \(w=\cos (x-y)+\ln (x+y),\) show that \(\frac{\partial^{2} w}{\partial x^{2}}-\frac{\partial^{2} w}{\partial y^{2}}=0\)

If \(w=f(x, y),\) where \(x=r \cos \theta\) and \(y=r \sin \theta,\) show that \(\left(\frac{\partial w}{{\partial }{} x}\right)^{2}+\left(\frac{\partial w}{\partial y}\right)^{2}=\left(\frac{\partial w}{\partial r}\right)^{2}+\frac{1}{r^{2}}\left(\frac{{\partial w} }{\partial \theta}\right)^{2}\)