다중적분에서의 치환

Multiple integration is a fundamental tool for calculating physical quantities over a region, such as the total mass of an elliptical plate.

The equation of an ellipse defines the boundary of this region. To find the total mass, a density function must be integrated over every point inside this boundary.

In standard rectangular coordinates, this boundary creates complicated limits involving complex square root functions, which are difficult to solve.

To simplify the calculation, a change of variables is used. By substituting x = au and y = bv, the elliptical boundary is transformed into a unit circle in the uv-plane.

This transformation deforms the area, so the integral is rewritten using variables u and v, incorporating the Jacobian found from the change of coordinates.

Next, to further simplify the integration over the circular region, the variables are converted to polar coordinates.

This step introduces a second Jacobian factor, r, which accounts for the geometry of circular sectors.

By integrating the new function and incorporating both Jacobian correction factors, the elliptical problem is transformed into a straightforward calculation with constant limits.