Meaning of the Jacobian The Jacobian is a magnification (or reduction) factor
that relates the area of a small region near the point \((u, v)\) to the area of
the image of that region near the point \((x, y)\) a. Suppose \(S\) is a rectangle
in the \(u v\) -plane with vertices \(O(0,0)\) \(P(\Delta u, 0),(\Delta u, \Delta
v),\) and \(Q(0, \Delta v)\) (see figure). The image of \(S\) under the
transformation \(x=g(u, v), y=h(u, v)\) is a region \(R\) in the \(x y\) -plane. Let
\(O^{\prime}, P^{\prime},\) and \(Q^{\prime}\) be the images of
O, \(P,\) and \(Q,\) respectively, in the \(x y\) -plane, where \(O^{\prime},
P^{\prime},\) and
\(Q^{\prime}\) do not all lie on the same line. Explain why the coordinates
of \(\boldsymbol{O}^{\prime}, \boldsymbol{P}^{\prime},\) and \(Q^{\prime}\) are
\((g(0,0), h(0,0)),(g(\Delta u, 0), h(\Delta u, 0))\)
and \((g(0, \Delta v), h(0, \Delta v)),\) respectively.
b. Use a Taylor series in both variables to show that
$$\begin{array}{l}
g(\Delta u, 0) \approx g(0,0)+g_{u}(0,0) \Delta u \\
g(0, \Delta v) \approx g(0,0)+g_{v}(0,0) \Delta v \\
h(\Delta u, 0) \approx h(0,0)+h_{u}(0,0) \Delta u \\
h(0, \Delta v) \approx h(0,0)+h_{v}(0,0) \Delta v
\end{array}$$ where \(g_{u}(0,0)\) is \(\frac{\partial x}{\partial u}\) evaluated
at \((0,0),\) with similar meanings for \(g_{v}, h_{u},\) and \(h_{v}\)
c. Consider the vectors \(\overrightarrow{O^{\prime} P^{\prime}}\) and
\(\overrightarrow{O^{\prime} Q^{\prime}}\) and the parallelogram, two of whose
sides are \(\overrightarrow{O^{\prime} P^{\prime}}\) and
\(\overrightarrow{O^{\prime} Q^{\prime}}\). Use the cross product to show that
the area of the parallelogram is approximately \(|J(u, v)| \Delta u \Delta v\)
d. Explain why the ratio of the area of \(R\) to the area of \(S\) is
approximately \(|J(u, v)|\)
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