Theorem mathematic for relativity

[nulliusinverb]

hello!:

my problem is about of a theorem mathematic,as I prove the following theorem?

F(x)=F(a) + \sum^{n}_{i=1}(x^{i}-a^{i})H_{i}(x)

good first start with the fundamental theorem of calculus: (for proof):

F(x) - F(a) = \int^{x}_{a}F'(s)ds sustitution: s=t(x - a) + a \Rightarrow [a,x] to [0,1] then:
ds=dt(x - a) later:
f(x) - F(a)= (x - a)\int^{1}_{0}F'(t(x - a) +a)dt

okk my problem is how to get to the sum \sum?

is physics relativistic forum, because of this theorem I can get to the change of coordinates in the Einstein equations and find bases for the manifolds of space-time. thanks!

 

[mathman]

http://en.wikipedia.org/wiki/Taylor_series

Your representation is a modification of the standard Taylor series representation, where H_i(a) (not H_i(x)) is related to the i^th derivative of F.

Furthermore the term in parenthesis should be (x-a)ⁱ not xⁱ - aⁱ

 

[Fredrik]

Furthermore the term in parenthesis should be (x-a)ⁱ not xⁱ - aⁱ

I know it's not clear from what he said, but he has a different theorem in mind. Wald's statement of the theorem goes like this:

If $F:\mathbb R^n\to\mathbb R$ is $C^\infty$, then for each $a=(a^1,\dots,a^n)\in\mathbb R^n$ there exist $C^\infty$ functions $H_\mu$ such that for all $x\in\mathbb R^n$ we have
$$F(x)=F(a)+\sum_{\mu=1}^n(x^\mu-a^\mu)H_\mu(x).$$ Furthermore, we have
$$H_\mu(a)=\frac{\partial F}{\partial x^\mu}\bigg|_{x=a}.$$​

A similar theorem is stated and proved in Isham's book on differential geometry, page 82. So nulliusinverb, I suggest you take a look at that.

 

[nulliusinverb]

Fredrick thank you very much, the book is recommended to study this issue, although the theorem is raised from other values ​​is the same. thank!

ps: "Modern Differential Geometry for Physicists" autor: Isham

 

[mathman]

Notation is a major problem here. xⁱ usually means the i^th power of x. For this theorem it means the i^th component of a vector x ε R_n.

I suggest that, when anyone asks a question, make sure the notation is clear!

 

[nulliusinverb]

i apologize for the delay, here is the proof:

F:ℝ^{n}\rightarrowℝ

i have:

F(\vec{x})-F(\vec{a})= \sum^{m}_{μ=1}F(t(x^{μ}-a^{μ})+a^{μ},0...,0)^{t=1}_{t=0}
then:
=\sum^{m}_{μ=1}(x^{μ}-a^{μ})\int^{1}_{0}\frac{\partial F}{\partial u^{μ}}((t(x^{μ}-a^{μ})+a^{μ},0...,0)dt
where:

H_{μ}(\vec{x})=\int^{1}_{0}\frac{\partial F(\vec{x})}{\partial u^{μ}}dt

finally:

F(\vec{x})-F(\vec{a})= \sum^{m}_{μ=1}H_{μ}(\vec{x})(x^{μ}-a^{μ})

qed

thanks you very much to all!