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:
$$\frac{1}{l!}...
Thank you for your answer. So I got it right, that it is possible to interpret the differential via the chain rule as $$(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...
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...
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...