# Taylor expansion, of gradient of a function, in multiple dimensions

1. Sep 11, 2011

### Whenry

Hello all,

I understand that the taylor expansion for a multidimensional function can be written as

$f(\overline{X}$ + $\overline{P}$) = $f(\overline{X}) + \nabla f(\overline{X}+t\overline{P})(\overline{P})$

where t is on (0,1).

Although I havent seen that form before, it makes sense.

But I don't understand the integral in the following the Taylor expansion,

$\nabla f(\overline{X}$ + $\overline{P}$) = $\nabla f(\overline{X}) + \int^{1}_{0} \nabla^{2} f(\overline{X}+t\overline{P})(\overline{P})dt$

Could someone help me understand the derivation?

Thank you,

Will

Last edited: Sep 11, 2011
2. Sep 17, 2011

### Stephen Tashi

I don't understand it. What kind of multiplication is going on in the last term? It appears to be two vectors multiplied together. Is it a dot product?

I don't either, but this is an interesting formula and I would like to know where you saw it. Is this from a subject like fluid dynamics? Can you give a link to a page?

3. Sep 17, 2011

### Whenry

4. Sep 17, 2011