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

[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 haven't 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

 

[Stephen Tashi]

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).

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?

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

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

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?

 

[Whenry]

Regarding the first question - I apologize, I should I put a transpose on the gradient vector: delta_f(X+P)^T * P

Regarding the seconde question, you can find the lecture here, it is on slide 10.

http://terminus.sdsu.edu/SDSU/Math693a_s2004/Lectures/02/lecture.pdf

 

[Stephen Tashi]

I found something like it in the Wikipedia article on the Mean Value Theorem: http://en.wikipedia.org/wiki/Mean_value_theorem

It's in the section "Mean value theorem for vector-valued functions".