# Calculus - Differentials and Partial Derivatives

1. Jun 22, 2012

### Combinatorics

1. The problem statement, all variables and given/known data

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 $x$, $y$, $dx$, $dy$:
$du= F(x,y,dx,dy)=u_xdx +u_ydy$.

2. Relevant equations
3. The attempt at a solution
I'll be glad to receive some guidance regarding the following:
1) Does the second order differential is assumed to be:
$d^2 f = ( \frac{ \partial}{ \partial x_1} \Delta x_1 +\frac{ \partial}{ \partial x_2} \Delta x_2 ) ^n f$ ?
2) If so, how should I solve this question?

Thanks in advance !

2. Jun 25, 2012

### bigplanet401

You know what du is, so what's $$d^2 u = d(du) = ?$$

3. Jun 25, 2012

### HallsofIvy

Staff Emeritus
df= fxdx+ fydy.

As bigplanet401 said, you want the differential of that- but you also need to know that differentials are skew commutative. That is, that dxdy= -dydx.

4. Jun 25, 2012

### Combinatorics

Thanks a lot !

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