- #1
mathmonkey
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Homework Statement
Let ##A## be open in ##\mathbb{R}^n##; let ##\omega## be a k-1 form in ##A##. Given ##v_1,...,v_k \in \mathbb{R}^n##, define
##h(x) = d\omega(x)((x;v_1),...,(x;v_k)),##
##g_j(x) = \omega (x)((x;v_1),...,\widehat{(x;v_j)},...,(x;v_k)),##
where ##\hat{a}## means that the component ##a## is to be omitted.
Prove that ##h(x) = \sum _{j=1}^k (-1)^{j-1} Dg_j (x) \cdot v_j . ##
Homework Equations
The problem is broken into 3 parts:
(a) Let ##X = \begin{bmatrix} v_1 ... v_k \end{bmatrix}##. For each ##j## let ##Y_j = \begin{bmatrix}v_1 ... \hat{v}_j ... v_k \end{bmatrix}##. Given ##(i, i_1,...,i_{k-1})##, show that
##detX(i,i_1,...,i_{k-1}) = \sum _{j=1}^k (-1)^{j-1}v_{ij}detY_j(i_1,...,i_{k-1}).##
(b) Verify the theorem in the case ##\omega = fdx_I##.
(c) Complete the proof.
The Attempt at a Solution
I'm stuck on part (b), however. By the definition given in the text, if ##\omega = fdx_I## then ##d\omega = df \wedge dx_I##. I'm not quite sure how to link the result of part (a) to prove part (b). If anyone can shed any light on this problem I'd be really grateful! Thanks.