What is the Jacobian value for a change of variable in the integral f(x+{y})?

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SUMMARY

The Jacobian value for the change of variable in the integral f(x+{y}) is determined to be -2. This is derived from the Jacobian matrix, which consists of partial derivatives based on the conditions of y being positive or negative. The matrix is structured as | 1 1 | | 1 -1 |, leading to a determinant of -2. This analysis confirms that the integral can be evaluated by splitting the region of integration based on the sign of y.

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eljose
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let be the integral:

f(x+{y}) where {y}=y iff y>0 and -y iff y<0, then we make the change of variable x+{y}=u x=v then what would be the value of Jacobian?
 
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|y| isn't differentiable so one cannot do it directly. but one may split the region of integration into the regions where y is positive and negative and thence do the integral.
 


The Jacobian value for a change of variable in an integral is the determinant of the Jacobian matrix, which is a matrix of partial derivatives. In this case, the Jacobian matrix would have two rows and two columns, with the first row containing the partial derivatives of u with respect to x and y, and the second row containing the partial derivatives of v with respect to x and y.

Since we are making the change of variable x+{y}=u, the partial derivative of u with respect to y would be 1 if y>0 and -1 if y<0. Similarly, the partial derivative of v with respect to y would be 1 if y>0 and -1 if y<0.

Therefore, the Jacobian matrix would be:
| 1 1 |
| 1 -1 |

The determinant of this matrix is -2, so the Jacobian value for this change of variable would be -2.
 

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