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Changing Coordinates

Very often we encounter functions that are more naturally expressed in coordinates other than Cartesian (x,y) due to symmetry. Sometimes, even if the function is ``nice'' in Cartesian coordinates, the boundaries of the region of integration can be more easily expressed in another system of coordinates. In this chapter, we'll start with (hopefully familiar) polar coordinates and integrate functions of two variables in these coordinates. Then we'll move into the third dimension and discuss volume integrals in two common coordinate systems. We'll finish off with some general statements about changing coordinates. When changing coordinates for integration, there are generally three things that need to be done:

1.
rewrite the function in the new coordinates,
2.
describe the region of integration in the new coordinates, and
3.
determine the area or volume element in the new coordinates.


 

Vector Calculus
8/20/1998