# Calculus- Differentials and multivariable functions

Find a differential of second order of a function $u=f(x,y)$ with continuous partial derivatives up to third order at least. Hint: Take a look at $du$ as a function of the variables $x$, $y$, $dx$, $dy$: $du= F(x,y,dx,dy)=u_xdx +u_ydy$.

Can someone please explain me what should I do in this question?

The question is stated in a rather old-fashioned language. The "second differential" is probably the second derivative. It should be well-known that the second derivative is represented (in local coordinates) by the Hessian matrix $$f''(x,y) = \begin{pmatrix} \frac{\partial^2 u}{\partial x^2} &\frac{\partial^2 u}{\partial x \partial y} \\ \frac{\partial^2 u}{\partial y \partial x} &\frac{\partial^2 u}{\partial y^2} \end{pmatrix}.$$ Its action is therefore $$\begin{pmatrix} dx &dy \end{pmatrix}^T \begin{pmatrix} \frac{\partial^2 u}{\partial x^2} &\frac{\partial^2 u}{\partial x \partial y} \\ \frac{\partial^2 u}{\partial y \partial x} &\frac{\partial^2 u}{\partial y^2} \end{pmatrix} \begin{pmatrix} dx \\ dy \end{pmatrix}$$