Recent content by Maxi1995

Well I weren't sure how to cope with the $$\sigma.$$ My idea was to say that we find for every permutation a counter part, that deviates only by one transposition from our permutation, so to say its form might be $$\sigma \circ \tau,$$ wehere tau is a transposition. By the alternation of the...

Hello, we defined the wedge-product as follows Alt is the Alternator and the argument of Alt is the Tensor poduct of one k-form and a l-form (in this order w and eta). Suppose we have the wedge product of a 0-form (a smooth function) and a l-form , so the following may result: $$\frac{1}{l!}...

Thank you very much. :bow:

Thank you for your answer. So I got it right, that it is possible to interpret the differential via the chain rule as $$(f \circ\alpha)'(0)=df(\alpha(0))*\alpha'(0)=df(p)v?$$

We define the differential of a function f in $$p \in M$$, where M is a submanifold as follows In this case we have a smooth curve ans and interval I $$\alpha: I \rightarrow M;\\ \alpha(0)= p \wedge \alpha'(0)=v$$. How can I get that derivative at the end by using the definitions of the...

Hello, let $$M^n \subset \mathbb{R}^N$$ $$N^k \subset \mathbb{R}^K$$ be two submanifolds. We say a function $$f : M \rightarrow N$$ is differentiable if and only if for every map $$(U,\varphi)$$ of M the transformation $$f \circ \varphi^{-1}: \varphi(U) \subset \mathbb{R}^N \rightarrow...

Thank you very much for your answer, I got it.

Hello, let us define the Alternator $$Alt(T)$$ where T is a multilinear function $$Alt(T):= \frac{1}{k!} \sum_{\sigma \in S_n} sgn(\sigma) T (v_{\sigma(1)},...,v_{\sigma(k)}))$$. Further recognize that $$S_n$$ is the group of permutations and sgn the signum of the permutation. May...

Hello, I have another question in this context. Let us say we want to talk about the function as object, thus to say the function itself as element of $$\Lambda ^k V.$$ We derived the coefficients in our proofs by using the smooth k-form itself. We set the coefficients as...

Thank you for yor answers, I'm going to think about it and give a sign in case of further needed help. :)

Hello, thank you for your answer, I'm going to read through these pages. I'll give a sign in case of further questions.

Hello, how do I have to start to prove that cuboids are measurable in the context of the Lebesgue measure? Best wishes Maxi

Hello, thank you for your answer. @ fresh_42 What is f in your notation? Is it a curve? I understand what you said about the elements of the basis, thus to say that they are just scalars. But I found in a Wikipedia article (https://en.wikipedia.org/wiki/Differential_form#Differential_calculus)...

Hello, We defined a k-form on a smooth manifold M as a transfromation Where the right space is the one of the alternating k-linear forms over the tangent space in p. If we suppose we know, that we get a basis of this space by using the wedge-product and a basis of the dual space, then we might...