Problems

In Problems 1-12, find all the second partial derivatives.

In Problems 13-16, find all the third partials.

In Problems 19-24, find ∂²z/∂x², ∂²z/∂y², and ∂²z/∂x ∂y.

In Problems 25-28 find ∂²z/∂s², ∂²z/∂t², and ∂²z/∂s ∂t.

29            Suppose z = f(x + at) + g(x - at) where f and g have continuous second derivatives. Show that z satisfies the wave equation

30            Show that if

z = Ax² + Bxy + Cy² + Dx + Ey + F

then all the second partial derivatives of z are constant.

31             Let f(x, y) be the function

Find the first and second partial derivatives of f. Show that

(a)    ∂²f/∂x ∂y + ∂²f/∂y ∂x at (0,0),

(b)    ∂²f/∂x ∂y is not continuous at (0, 0).

Hint:

All the derivatives must be computed separately for the cases (x, y) = (0,0) and (x,y) ≠ (0,0).