- #1
theneedtoknow
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Clarification of "change of variables" for multiple integration
This isn't really a question about a specific math problem, but rather for the change of variables of multiple integration as a whole. When you change variables you have to multiply the new expression by the jacobian of the new functions you chose. So, if the determinant is a positive constant, you can just bring it outside of the integral signs. However, in my book, whenever the determinant is a negative constant, they bring out its positive reciprocal instead (For example, if the jacobian determinant is -1/3, they bring out a 3 instead)
Is this the proper way to do it?
For example, there is q eustion where you use the substituion u = x+y, v = x-y
the determinant of this is -2 ( ad - bc = -1 - 1 = 2), so in the solutions in the back of the book they've turned they've multiplied the new expression by 1/2 instead of by -2
Is this the rule? If determinant is a positive constant, you use that constant, and if it's a negative one, you use its positive reciprocal?
This isn't really a question about a specific math problem, but rather for the change of variables of multiple integration as a whole. When you change variables you have to multiply the new expression by the jacobian of the new functions you chose. So, if the determinant is a positive constant, you can just bring it outside of the integral signs. However, in my book, whenever the determinant is a negative constant, they bring out its positive reciprocal instead (For example, if the jacobian determinant is -1/3, they bring out a 3 instead)
Is this the proper way to do it?
For example, there is q eustion where you use the substituion u = x+y, v = x-y
the determinant of this is -2 ( ad - bc = -1 - 1 = 2), so in the solutions in the back of the book they've turned they've multiplied the new expression by 1/2 instead of by -2
Is this the rule? If determinant is a positive constant, you use that constant, and if it's a negative one, you use its positive reciprocal?