if we want to evaluate the integration $$I=\int\int(x^3y^3)(x^2+y^2)dA$$ over the region bounded by the curves $$xy=1,\\xy=3,\\x^2-y^2=1,\\x^2-y^2=4$$ I used the transformation $$u=xy,\\v=x^2-y^2$$ I found that the jacobian will be $$J=\frac{1}{-2y^2-2x^2}$$ Do I have to get the absolute value of the jacobian? If I did not take the absolute value , I will get the result of the integration = - 30
if I take the absolute value $$J=\frac{1}{2y^2+2x^2}$$ I will get the result= + 30 .
My friend told me we take absolute value of the jacobian only if it is a number .. if this is right .. why we do not take the absolute value if the jacobian is a function?..I think we are sure here that the jacobian is negative since we have x and y squared , so we have to take the absolute value!
Another question, if we have to take always the absolute value of the jacobian (whether it is a number or function) :
if the jacobian is for example $$J=-2x+y$$ It will be positive for some values of x and y only ! .. how can we apply the absolute value inside the double integration?
If the Jacobian is negative, then the orientation of the region of integration gets flipped.
You have to take the absolute value ALWAYS.