Double integrals
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Change of variables
For a function of one variable, the formula for the change of variable from x to u is [Equation goes here - download the original pdf to see it.] where [Equation goes here - download the original pdf to see it.] is a continuous function with a continuous derivative in the interval [Equation goes here - download the original pdf to see it.] and [Equation goes here - download the original pdf to see it.]. For a double integral the formula for the change of variables is [Equation goes here - download the original pdf to see it.] where J is the Jacobian, given by [Equation goes here - download the original pdf to see it.] This assumes that the functions [Equation goes here - download the original pdf to see it.] are continuous and have continuous partial derivatives in the region R* in the uv-plane; and that the point [Equation goes here - download the original pdf to see it.] corresponding to the point [Equation goes here - download the original pdf to see it.] in R* lies in R. Also for every point [Equation goes here - download the original pdf to see it.] in R there corresponds one and only one point [Equation goes here - download the original pdf to see it.] in R*. Furthermore, the Jacobian J is either positive throughout R* or negative throughout R*.
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Contents of
Double integrals
1 Double integral
2 Evaluation of a double integral
3 Some applications double integrals
4 Change of variables
5 The double integral in polar coordinates
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