SOLUTION (a) The Chain Rule gives \[ \frac{\partial z}{\partial r} = \frac{\partial
z}{\partial x} \frac{\partial x}{\partial r} + \frac{\partial
z}{\partial y} \frac{\partial y}{\partial r} = \frac{\partial
z}{\partial x}(2r) + \frac{\partial z}{\partial y}(2s) \] (b)
Applying the Product Rule to the expression in part (a), we get \[ \frac{\partial^2 z}{\partial r^2} =
\frac{\partial}{\partial r} \left( 2r \frac{\partial z}{\partial x} + 2s
\frac{\partial z}{\partial y} \right) \] \[ = 2 \frac{\partial z}{\partial x} + 2r
\frac{\partial}{\partial r} \left( \frac{\partial z}{\partial x} \right)
+ 2s \frac{\partial}{\partial r} \left( \frac{\partial z}{\partial y}
\right) \] But, using the Chain Rule again (see Figure 5), we
have \[ \frac{\partial}{\partial r} \left(
\frac{\partial z}{\partial x} \right) = \frac{\partial}{\partial x}
\left( \frac{\partial z}{\partial x} \right) \frac{\partial x}{\partial
r} + \frac{\partial}{\partial y} \left( \frac{\partial z}{\partial x}
\right) \frac{\partial y}{\partial r} = \frac{\partial^2 z}{\partial
x^2}(2r) + \frac{\partial^2 z}{\partial y \partial x}(2s) \]
\[ \frac{\partial}{\partial r} \left(
\frac{\partial z}{\partial y} \right) = \frac{\partial}{\partial x}
\left( \frac{\partial z}{\partial y} \right) \frac{\partial x}{\partial
r} + \frac{\partial}{\partial y} \left( \frac{\partial z}{\partial y}
\right) \frac{\partial y}{\partial r} = \frac{\partial^2 z}{\partial x
\partial y}(2r) + \frac{\partial^2 z}{\partial y^2}(2s) \]
Putting these expressions into Equation 5 and using the equality of the
mixed second-order derivatives, we obtain \[
\frac{\partial^2 z}{\partial r^2} = 2 \frac{\partial z}{\partial x} + 2r
\left( 2r \frac{\partial^2 z}{\partial x^2} + 2s \frac{\partial^2
z}{\partial y \partial x} \right) + 2s \left( 2r \frac{\partial^2
z}{\partial x \partial y} + 2s \frac{\partial^2 z}{\partial y^2} \right)
\] \[ = 2 \frac{\partial z}{\partial
x} + 4r^2 \frac{\partial^2 z}{\partial x^2} + 8rs \frac{\partial^2
z}{\partial x \partial y} + 4s^2 \frac{\partial^2 z}{\partial y^2}
\]
