Jacobian of u=y-x/1+xy and v=tan-1y - tan-1x . Find del(u,v)/del(x,y)

The Jacobian matrix of a vector-valued function is a matrix that contains all of the first order partial derivatives of the function. In this case, we have the functions u = y - x/(1 + xy) and v = arctan(y) - arctan(x).

The Jacobian matrix J of these functions with respect to x and y is given by:

J = [du/dx, du/dy] [dv/dx, dv/dy]

We need to compute each of these partial derivatives.

1. Compute du/dx:

du/dx = -1/(1 + xy) + x*(-y)/((1 + xy)^2)

2. Compute du/dy:

du/dy = 1 - x^2/((1 + xy)^2)

3. Compute dv/dx:

dv/dx = -1/(1 + x^2)

4. Compute dv/dy:

dv/dy = 1/(1 + y^2)

So, the Jacobian matrix is:

J = [-1/(1 + xy) + x*(-y)/((1 + xy)^2), 1 - x^2/((1 + xy)^2)] [-1/(1 + x^2), 1/(1 + y^2)]