Guichard 16.3.5

TheMeansOfProduction

In Guichard 16.3.5, we are asked to ``find an $f$ so that $\nabla f=\left<y\cos x,\sin x\right>$, or explain why there is no such $f$."

We know such an $f$ exists since $$\frac{\partial}{\partial y}(y\cos x)=\cos x=\frac{\partial}{\partial x}\sin x.$$ I'll leave it to someone else to find $f$.

NegativeOneTwelfth

ez pz $$f=y\sin x$$ because $$ \frac{\partial}{\partial{x}}y\sin x = y\cos x , \frac{\partial}{\partial{y}}y\sin x=\sin x $$ $$ \Rightarrow \nabla{f}=\langle y\cos x,\sin x \rangle $$