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

[eljose]

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?

 

[matt grime]

|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.

 

[M98Ranger]

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.