Help! Function of two variable

Question

Let w=f(x,y) be a function of two variables and let x=u+v, y=u-v. Show that ∂²w/∂u∂v=∂²w/∂x²-∂²w/∂y².

Answer 1

This requires you to compute both sides of the equation separately and then check that they are equal.

 

The left side asks for the second partial derivative of w with respect to v, then u.

 

∂w/∂u = ∂w/∂x ∂x/∂u + ∂w/∂y ∂y/∂u = ∂w/∂x + ∂w/∂y . (Both of the partials of x and y w.r.t u are equal to 1.)

 

Next, you must find the partial derivative of THIS w.r.t v, again using the Chain Rule.

 

∂²w/∂v∂u = ∂/∂u (∂w/∂x + ∂w/∂y) = ∂²w/∂x² ∂x/∂v + ∂²w/∂y∂x ∂y/∂v + ∂²w/∂x∂y ∂x/∂v + ∂²w/∂y² ∂y/∂v

 

               = ∂²w/∂x² – ∂²w/∂y∂x + ∂²w/∂x∂y – ∂²w/∂y²

 

               = ∂²w/∂x² – ∂²w/∂y² (the two middle terms cancel out!)

 

Since I have already simplified the left side to match the right side, no further work is necessary and the problem is finished.