Jacobian of x=e^u cosv and y=e^u sinv . Find del(u,v)/del(x,y)

To find the Jacobian of the transformation, we first need to find the partial derivatives of u and v with respect to x and y.

The given transformations are: x = e^u cos(v) y = e^u sin(v)

Differentiating these with respect to u and v, we get:

∂x/∂u = e^u cos(v) ∂x/∂v = -e^u sin(v) ∂y/∂u = e^u sin(v) ∂y/∂v = e^u cos(v)

The Jacobian matrix J of the transformation is then given by:

J = | ∂x/∂u ∂x/∂v | | ∂y/∂u ∂y/∂v |

J = | e^u cos(v) -e^u sin(v) | | e^u sin(v) e^u cos(v) |

The determinant of J, denoted as |J|, is then:

|J| = (e^u cos(v))^2 + (e^u sin(v))^2 = e^(2u) (cos^2(v) + sin^2(v)) = e^(2u)

The Jacobian of the inverse transformation, denoted as del(u,v)/del(x,y), is then the reciprocal of |J|, which is:

del(u,v)/del(x,y) = 1/|J| = 1/e^(2u)