In one line
The Jacobian \(J=\dfrac{\partial(u,v)}{\partial(x,y)}\) is the determinant of all first partial derivatives of \(u,v\) with respect to \(x,y\); it measures how a change of variables stretches area (or volume).
Explained simply
Take a small square on a rubber sheet and stretch the sheet. The square becomes a small parallelogram with a different area. The Jacobian is the factor by which areas are multiplied at that spot. In polar coordinates, a small patch \(dr\,d\theta\) has area \(r\,dr\,d\theta\), and that extra \(r\) is exactly the Jacobian.
Technically, for \(u=u(x,y)\), \(v=v(x,y)\): \(J=\dfrac{\partial(u,v)}{\partial(x,y)}=\begin{vmatrix}u_x&u_y\\v_x&v_y\end{vmatrix}\). For three functions \(u,v,w\) of \(x,y,z\) it is the \(3\times3\) determinant of their partial derivatives. This is a self-learning topic.
Where it is used: in double and triple integrals, when you change variables (for example to polar, cylindrical or spherical coordinates), the area or volume element must be multiplied by the absolute value of the Jacobian. A zero Jacobian also warns that the new variables are not independent of each other.
How it works
- Write the functions to be differentiated (top letters) and the variables (bottom letters).
- Place the partial derivatives row by row: row 1 for \(u\), row 2 for \(v\), and so on.
- Evaluate the determinant.
- If \(J=\dfrac{\partial(u,v)}{\partial(x,y)}\) and \(J'=\dfrac{\partial(x,y)}{\partial(u,v)}\), then \(JJ'=1\).
Key formula
\(\dfrac{\partial(u,v)}{\partial(x,y)}=u_xv_y-u_yv_x\); \(\ \dfrac{\partial(u,v,w)}{\partial(x,y,z)}=\begin{vmatrix}u_x&u_y&u_z\\v_x&v_y&v_z\\w_x&w_y&w_z\end{vmatrix}\); \(\ JJ'=1\).
Quick example
Polar coordinates: \(x=r\cos\theta\), \(y=r\sin\theta\).
\(\dfrac{\partial(x,y)}{\partial(r,\theta)}=\begin{vmatrix}\cos\theta&-r\sin\theta\\\sin\theta&r\cos\theta\end{vmatrix}=r\cos^2\theta+r\sin^2\theta=r\).
So \(\dfrac{\partial(r,\theta)}{\partial(x,y)}=\dfrac1r\) by \(JJ'=1\).
Three variables: \(u=x+y+z\), \(v=y+z\), \(w=z\). \(J=\begin{vmatrix}1&1&1\\0&1&1\\0&0&1\end{vmatrix}=1\). Spherical coordinates give the well-known \(r^2\sin\theta\).
Common mistakes
- Mixing the layout: each row must belong to one function and each column to one variable, consistently.
- Confusing \(\dfrac{\partial(u,v)}{\partial(x,y)}\) with its reciprocal.
- Sign errors in expanding a \(3\times3\) determinant.
How MU asks it
As a self-learning topic expect a 3–5-mark part: "If \(x=u(1-v)\), \(y=uv\), find \(\dfrac{\partial(x,y)}{\partial(u,v)}\)" (answer \(u\)) or "Verify \(JJ'=1\) for polar coordinates". Write the determinant layout, each partial derivative, and the expanded value.