1. Definition
For a map (u,v)→(x,y), the Jacobian J=∂(x,y)/∂(u,v)=x_u y_v−x_v y_u. Under change of variables, dxdy=|J|dudv.
2. Symbols
| x_u,x_v | partials of x |
| y_u,y_v | partials of y |
| J | Jacobian |
| |J| | area scale |
3. How it works
- Find four partials numerically;
- Apply the cross-product formula;
- |J| scales area.
4. Steps
- Enter x(u,v), y(u,v);
- Enter the point;
- Click Calculate.
5. Example
Example: x=u+v, y=u²−v at u=1,v=2.
Solution: x_u=1,x_v=1,y_u=2,y_v=−1; J=−3, |J|=3.
6. Pitfalls
Use |J| in integrals;
J=0 means degeneracy;
Partials are w.r.t. new variables.