1.12 Jacobians
Definition 1.12.1. If \(F(u,v)\) and \(G(u,v)\) are differentiable in a region, the Jacobian of \(F\) and \(G\) with respect to \(u\) and \(v\) is the determinant \[ \frac {\partial (F,G)}{\partial (u,v)} = \begin {vmatrix} \frac {\partial F}{\partial u} & \frac {\partial F}{\partial v}\\[1ex] \frac {\partial G}{\partial u} & \frac {\partial G}{\partial v} \end {vmatrix}. \] Similarly, for three functions of three variables, the third-order determinant \[ \frac {\partial (F,G, H)}{\partial (u,v,w)} = \begin {vmatrix} \frac {\partial F}{\partial u} & \frac {\partial F}{\partial v} & \frac {\partial F}{\partial w}\\[1ex] \frac {\partial G}{\partial u} & \frac {\partial G}{\partial v} & \frac {\partial G}{\partial w}\\[1ex] \frac {\partial H}{\partial u} & \frac {\partial H}{\partial v} & \frac {\partial H}{\partial w} \end {vmatrix} \] is the Jacobian of \(F\), \(G\), \(H\) with respect to \(u\), \(v\), \(w\).
Properties of Jacobians
Both properties follow from the determinant form of the general chain rule.
- 1.
- Chain rule. If \(x\) and \(y\) are functions of \(u\) and \(v\), while \(u\) and \(v\) are functions of \(r\) and \(s\), then \[\frac {\partial (x,y)}{\partial (r,s)} = \frac {\partial (x,y)}{\partial (u,v)}\,\frac {\partial (u,v)}{\partial (r,s)} .\]
- 2.
- Inverse. If \(u\) and \(v\) are functions of \(x\) and \(y\), and the equations can be solved for \(x\) and \(y\) in terms of \(u\) and \(v\), then \[\frac {\partial (u,v)}{\partial (x,y)}\,\frac {\partial (x,y)}{\partial (u,v)}=1 .\] This is property 1 with \((r,s) = (u,v)\): the composite of a mapping and its inverse is the identity, whose Jacobian is \(1\).
Example 1.12.2. If \(x = uv\) and \(y =\frac {u + v}{u -v}\), find \(\frac {\partial (u,v)}{\partial (x,y)}\).
Solution. Computing \(u_x\), \(u_y\), \(v_x\) and \(v_y\) directly would require solving for \(u\) and \(v\), which is awkward. The derivatives of \(x\) and \(y\) are easy, so compute the inverse Jacobian first: \[\frac {\partial (x,y)}{\partial (u,v)} = \begin {vmatrix} v & u\\[1ex] -\frac {2v}{(u-v)^2} & \frac {2u}{(u-v)^2} \end {vmatrix} = \frac {2uv}{(u-v)^2} + \frac {2uv}{(u-v)^2} = \frac {4uv}{(u -v)^2}.\] By property 2, \[\frac {\partial (u,v)}{\partial (x,y)} = \frac {(u -v)^2}{4uv},\] wherever \(uv\neq 0\).
Exercise 1.12.3. If \(u = xyz\), \(v = x^2 + y^2 + z^2\) and \(w = x+y+z\), find \(\frac {\partial (x,y,z)}{\partial (u,v,w)}\).
Answer: \(\frac {\partial (u,v,w)}{\partial (x,y,z)} = 2(x-y)(y-z)(x-z)\), so \(\frac {\partial (x,y,z)}{\partial (u,v,w)} = \frac {1}{2(x-y)(y-z)(x-z)}\) wherever \(x\), \(y\), \(z\) are distinct.
Example 1.12.4. Find the Jacobian \(\frac {\partial (u,v)}{\partial (r,\theta )}\), where \(u = x^2 - y^2\), \(v = 2xy\) and \(x = r\cos \theta \), \(y = r\sin \theta \).
Solution. By property 1, \begin {align*} \frac {\partial (u,v)}{\partial (r,\theta )} & = \frac {\partial (u,v)}{\partial (x,y)}\,\frac {\partial (x,y)}{\partial (r,\theta )} = \begin {vmatrix} 2x & -2y\\ 2y & 2x \end {vmatrix}\, \begin {vmatrix} \cos \theta & -r\sin \theta \\ \sin \theta & r\cos \theta \end {vmatrix}\\[1ex] & = (4x^2 + 4y^2)\cdot r = 4r^2\cdot r = 4r^3 . \end {align*}
As a check, \(u = r^2\cos 2\theta \) and \(v = r^2\sin 2\theta \) directly, and \(\begin {vmatrix} 2r\cos 2\theta & -2r^2\sin 2\theta \\ 2r\sin 2\theta & 2r^2\cos 2\theta \end {vmatrix} = 4r^3\).
Example 1.12.5. Interpret the Jacobian of the linear mapping \(u = 3x + y\), \(v = x - 2y\) geometrically.
Solution. \[\frac {\partial (u,v)}{\partial (x,y)} = \begin {vmatrix}3 & 1\\ 1 & -2\end {vmatrix} = -7 .\] The unit square with sides \(\langle 1,0\rangle \) and \(\langle 0,1\rangle \) is sent to the parallelogram with sides \(\langle 3,1\rangle \) and \(\langle 1,-2\rangle \), whose area is \(|3(-2) - 1\cdot 1| = 7\). So the mapping multiplies areas by \(|J| = 7\), and the negative sign records that it reverses orientation: a counter-clockwise boundary becomes a clockwise one. For a non-linear mapping the same is true of small regions near each point, which is why \(|J|\) appears as the scale factor in the change of variables formula for double integrals.
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