Discussion
If \( u = x + y \) and \( v = x - y \), the Jacobian \( \partial(u,v)/\partial(x,y) \) equals
Answer: C
The four first derivatives are \( u_{x} = 1 \), \( u_{y} = 1 \), \( v_{x} = 1 \), \( v_{y} = -1 \), so
\[ \frac{\partial(u,v)}{\partial(x,y)} \;=\; \begin{vmatrix} 1 & 1 \\ 1 & -1 \end{vmatrix} \;=\; (1)(-1) - (1)(1) \;=\; -2. \]
The magnitude two is the area stretch of this rotation and scaling, and the sign records that the map reverses orientation
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