If and then the Jacobian of x, y, and z with respect to u, v, and w is . Find the Jacobian for the following change of variables:

Question

If  and  then the Jacobian of x, y, and z with respect to u, v, and w is  . Find the Jacobian for the following change of variables:

Quizplus · Accepted Answer

We can use the chain rule to find the Jacobian for the change of variables. Let

u = x 
v = y 
w = z + 1

Then, we can write the original variables as:

x = u 
y = v 
z = w - 1

Taking the partial derivatives, we have:

∂x/∂u = 1 
∂x/∂v = 0 
∂x/∂w = 0

∂y/∂u = 0 
∂y/∂v = 1 
∂y/∂w = 0

∂z/∂u = 0 
∂z/∂v = 0 
∂z/∂w = 1

Therefore, the Jacobian of x, y, and z with respect to u, v, and w is:

|1 0 0| 
|0 1 0| 
|0 0 1|

which is equivalent to 11eaa595_6272_7625_a696_d5cb25a31662_TB4584_11_TB4584_11.

If and Then the Jacobian of X, Y