Actually, the questioner is asking you to "discover" a well-known and completelyLHS said:Thanks :) It is annoying when they set questions where there just seems to be some arbitrary factorization which you can spend hours on and learn relatively little from. Have a good day!
A change of variables in a triple integral is a method used to simplify the integration process by transforming the original coordinate system into a new one. This is done by substituting new variables for the original ones in the integral, making it easier to evaluate.
A change of variables is useful when the original integral is difficult to solve or when the boundaries of the integral are complex. By choosing appropriate new variables, the integral can be transformed into a simpler form, making it easier to evaluate.
The first step is to identify the boundaries of the integral and determine if they are complex or difficult to work with. Then, choose new variables that will simplify the integral. These variables should be related to the original ones through a transformation equation. Finally, substitute the new variables into the integral and evaluate.
No, not all changes of variables can be used in a triple integral. The chosen variables must satisfy certain conditions such as being one-to-one and differentiable. Additionally, the Jacobian determinant must be nonzero for the transformation to be valid.
To determine if a change of variables is suitable for a triple integral, you can check if the boundaries of the integral become simpler or if the new integral is easier to evaluate. Additionally, you can calculate the Jacobian determinant and make sure it is nonzero, as this is a necessary condition for the transformation to be valid.