$.</li>
    </ol>
  </div>We now generalize to higher dimensions. If I want to write
  $\vec f(\vec g(\vec x))$, then $\vec x$ must be a vector in the
  domain of $g$. After computing $\vec g(\vec x)$, we must get a
  vector that is in the domain of $f$. Since the chain rule in
  first semester calculus states $(f(g(x))'=f'(g(x))g'(x)$, then in
  high dimension it should state $D(f(g(x)) = Df(g(x))Dg(x)$, the
  product of two matrices.

  <div class="problem">
    In <a href=
    "#differential-for-volume-o