Exercise 54

Assume that all the given functions have continuous second-order partial derivatives. Suppose \(z = f(x, y)\), where \(x = g(s, t)\) and \(y = h(s, t)\). (a) Show that \[ \frac{\partial^2 z}{\partial t^2} = \frac{\partial^2 z}{\partial x^2}\left(\frac{\partial x}{\partial t}\right)^2 + 2\frac{\partial^2 z}{\partial x \partial y}\frac{\partial x}{\partial t}\frac{\partial y}{\partial t} + \frac{\partial^2 z}{\partial y^2}\left(\frac{\partial y}{\partial t}\right)^2 + \frac{\partial z}{\partial x}\frac{\partial^2 x}{\partial t^2} + \frac{\partial z}{\partial y}\frac{\partial^2 y}{\partial t^2} \] (b) Find a similar formula for \(\partial^2 z/\partial s \partial t\).