# Tagged Questions

It is a quotient - of the tensor algebra, obtained by taking graded sum over whole numbers $n$ of $n$-fold tensor products - by the ideal generated by elements of the form $a\otimes a$. We write the residue class of $a\otimes b$ in this algebra, as $a\wedge b$ and call it the wedge product.

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### A natural Poisson bivector on the tangent bundle?

For a smooth manifold $M$, there is a natural $1$-form $\theta$ on $T^*M$ such that $\Bbb d \theta$ is a symplectic form. Somewhat symetrically, on $TM$ there is a natural tangent field $V$. Is it ...
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### Geometric Significance of some features of the Exterior Algebra

I've been tinkering with differential forms for a while now, and I've had a few questions all rolled into one trying to understand them. The exterior derivative is quite natural to me - it looks just ...
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### Restriction of an $n$-form to a submanifold

I am currently studying exterior calculus on manifolds and am not sure if I understand things correctly. In my textbook (R.W.R. Darling, Differential Forms and Connections) there is an example of a $2$...
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### Why is the wedge product of a 1-form and itself $0$? [closed]

Why is the wedge product of a 1-form and itself $0$? Why doesn't this apply to 2-forms?
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### Change of coordinates using the wedge product?

I while back (over a year ago) I was told about wedge products (or something very similar) and how they can be used to change e.g. the curl in Cartesian coordinates to in spherical coordinates. ...
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### Is this an alternate characterization of $\lambda$-rings? Or, what is like a $\lambda$-ring but for symmetric rather than exterior powers?

This is a question about $\lambda$-rings. A $\lambda$-ring is a commutative ring together with operations $\lambda^n$ for each whole number $n$ which are analogous to the $n$th exterior power and ...
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### Connecting the regular representation of $\mathfrak{so}(3)$ and the exterior algebra of $\mathbb{R}^3$

It is well known that the regular representation of $\mathfrak{so}(3)$ is the so-called "cross product" matrix $A(x)$ which follows $A(x)y = x\times y$, and $x,y\in\mathbb{R}^3$, while the cross ...
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### Equivalent conditions for $\mathfrak{F}$ to be a differential ideal

Heres the question: Let $\mathfrak{F}$ be an ideal of forms on a manifold $M$ locally generated by $r$ independent $1$-forms. Say $\mathfrak{F}$ is generated by $\omega_1,\ldots, \omega_r$ on $U$. ...
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### Exterior Derivative vs. Covariant Derivative vs. Lie Derivative

In differential geometry, there are several notions of differentiation, namely: Exterior Derivative, $d$ Covariant Derivative/Connection, $\nabla$ Lie Derivative, $\mathcal{L}$. I have listed them ...
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### Powers of two in basis vectors indexes for wedge products

I am reading Fundamentals of Grassmann Algebra for game developers and it dawned on me that a slight tweak in the notation would make things easier. Remember that for n dimensions that number of basis ...
### Prove $\det(T^*)=\det(T)$ using exterior algebta
Let $T:V\to V$ be a linear operator on a finite-dimensional vector space $V$. $T^*:V^*\to V^*$ is the transpose of $T$ (i.e., $T^*(\lambda)=\lambda\circ T$). I want to find a coordinate-free proof ...