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3

They both may be right. If wedge product differs (Similar problem) and we set the definition of exterior derivative as $$d\omega=\sum_{I}d\omega_I\wedge dx^I,$$ then $d$ may differs as well (cause $\wedge$ appears). If we take axiomatic approach to exterior derivative, then one of axioms says ...

2

I'll propose to you another (slightly different, but isomorphic) definition of the adjugate (classical adjoint). Im borrowing from section 8 of http://people.reed.edu/~jerry/332/27exterior.pdf . Let $f:V\rightarrow V$ (with $n$ the dimension of $V$). We have a canonical isomorphism $\phi:V=\wedge^1 V\rightarrow\mathrm{Hom}(\wedge^{n-1} V,\wedge^n V)$ ...

2

In general, consider a one form $\beta = \sum_i \beta_i dx^i$ on $U\subset\mathbb R^m$. If $$\| \beta\|^2 = \beta_1^2 + \cdots + \beta_m^2 \neq 0$$ on $U$, then the $(n-1)$-form $$\alpha = \frac{1}{\|\beta\|^2} \sum_i (-1)^{i-1} \beta_i dx^1 \wedge \cdots \wedge \hat{dx^i} \wedge \cdots \wedge dx^m$$ on $U$ safisfies $$\beta \wedge \alpha = dx^1 ... 1 We do, but for historical reasons they are called (linear) isometries. 1 Your derivation is right and equation [1] cannot be correct without further assumptions. In fact the exterior derivative of a 1-form can be shown to be $$\mathrm{d}\omega(X,Y)= X(\omega(Y)) - Y(\omega(X))-\omega([X,Y])$$ which is equivalent to what you wrote. 1 A very nice gentle (albeit abstract) introduction to forms and connections can be found in R.W.R Darling's Differential Forms and Connections (1), a more physics based text book would have to be Nakahara's Geometry, Topology & Physics (2) - these helped me greatly when I had a similar need to you. Good Luck! (1) ... 1 you can read S.S.Chern's"lectures on differential geometry". He did very well on that book 1 First we parametrice the surface as follows,$$\left\{\begin{matrix}x&=&x\\y&=&\frac{1}{x}\\z&=&z\end{matrix}\right.$$so we get that,$$\left\{\begin{matrix}dx&=&dx\\dy&=&-\frac{1}{x^2}dx\\dz&=&dz\end{matrix}\right.$$so in xy=1 it is,$$x dy\wedge dz+ydz\wedge dx=-\frac{1}{x}dx\wedge dz-\frac{1}{x}dz\wedge ...

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