The Jacobian of a transformation measures how the variables in the original coordinate system change when transformed into a new coordinate system. In this case, we have the transformation:
x = 6u - v
y = -5u - 5v
To find the Jacobian of this transformation, we need to calculate the partial derivatives of x with respect to u and v, and the partial derivatives of y with respect to u and v. The Jacobian matrix is then formed by arranging these partial derivatives in a specific order.
Let's calculate the partial derivatives first:
∂x/∂u = 6
∂x/∂v = -1
∂y/∂u = -5
∂y/∂v = -5
Now we can form the Jacobian matrix:
J = [∂x/∂u ∂x/∂v]
[∂y/∂u ∂y/∂v]
Substituting the partial derivatives we calculated earlier, we get:
J = [6 -1]
[-5 -5]
So, the Jacobian of the transformation x = 6u - v and y = -5u - 5v is:
J = [6 -1]
[-5 -5]