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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?

Thanks in advance!

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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 $$ \begin{pmatrix} dx &dy \end{pmatrix}^T \begin{pmatrix} \frac{\partial^2 u}{\partial x^2} &\frac{\partial^2 u}{\partial x \partial y} \\ \frac{\partial^2 u}{\partial y \partial x} &\frac{\partial^2 u}{\partial y^2} \end{pmatrix} \begin{pmatrix} dx \\ dy \end{pmatrix} $$

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Thanks ! But I still can't figure out what should I do... What does it mean to find a differential that satisfies the above conditions? Hope you'll be able to help me abit further Thanks ! – joshua Jun 17 '12 at 8:33
As I said, I think it's just the differential of the differential. – Siminore Jun 17 '12 at 9:48

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