Find the differential $dw$ of the function $w=f(x,y,z)=3x^{2}\sin ^{2}y\cos z$.

Solution The function $f$ is defined everywhere in space. We begin by finding the partial derivatives of $f.$ $$\begin{eqnarray*} f_{x}(x,y,z) &=&6x\;\sin^{2}\;\!y\;\cos\;z\qquad f_{y}(x,y,z) =6x^{2}\;\sin\;y\;\cos\;y\cos\;z\\ f_{z}(x,y,z) &=&-3x^{2}\sin^{2}\;\!y\;\sin\;z \end{eqnarray*}$$

847

Since the partial derivatives are continuous everywhere, we have $$\begin{eqnarray*} dw&=&f_{x}(x,y,z)\ dx+f_{y}(x,y,z)\ dy+f_{z}(x,y,z)\ dz\\ &=&6x\;\sin^{2}\;\!y\;\cos\;z\,dx+6x^{2}\;\sin\;y\;\cos\;y\;\cos\;z\,dy-3x^{2}\;\sin^{2}\;\!y\;\sin\;z\,dz \end{eqnarray*}$$