Замена переменных в кратных интегралах

In multivariable calculus, evaluating a multiple integral over a region, R, often seems difficult as the region, R, may have complex boundaries in x and y coordinates.

To simplify this mathematical evaluation, a change of variables is made into a new coordinate system defined by the variables u and v through a specific set of transformation equations.

This transformation completely reshapes the original region, R, from the xy-plane onto a new region, S, in the uv-plane, creating two different area elements, dxdy and dudv, in two different coordinate systems.

To link these two area elements, a critical component, the Jacobian, is calculated. It measures how much the original region, R, is stretched or compressed during the transformation to the new region, S.

The Jacobian is obtained by evaluating a determinant that contains the partial derivatives of the original variables x and y with respect to the new variables u and v.

Finally, the newly substituted function is multiplied by the absolute value of the Jacobian. This step completes the transformation process and yields the fully rewritten integral over the new region, S.