# Tagged Questions

For questions about differential forms which commonly arise in differential geometry, and sometimes in multivariable calculus.

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### Alternative to Arnold's mathematical methods

I have difficulties understanding Arnold's book of mathematical methods of classical mechanics. Yet I should get some familiarity with the subjects found at chapters 3,4,7,8 before next semester to ...
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### Multivariable calc “second course” that does differential forms

I've worked through a computation-heavy, "standard" but quite nonrigorous treatment of multivariable calculus in the past. What book would do well as a rigorous (but not overly) "second course"? In ...
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### Definition of integration of differential forms

I am trying to understand precisely the following paragraph: Question Why would he define the support $K$ of a form $\omega$ defined on an open set $U$ as a subset $K\subseteq M$ instead of a ...
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### What is the intuition behind differential forms?

I am comfortable with the way physicists use differentials as elements of area/volume. I know the (algebraic) formal definition of differential forms, but it makes no intuitive sense, especially since ...
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Let $M$ be a smooth manifold of dimension $n$ and let $V$ be a $\mathbb R$-vector space of finite dimension $\ell$. A $k$-form on $M$ with values on $V$ is a map $\omega$ on $M$ such that: $$\omega_p:=... 2answers 47 views ### Grammatically confused: \omega=4dV for 3-form \omega and volume in \Bbb R^4? Background: Against the advice I should have been given but wasn't, I'm taking a Lie theory course with no background in differential geometry. We finally made it into the part of the course where we ... 1answer 69 views ### Volume Forms Induced by Embedding Let (M, g) be a Riemannian Manifold of dimension d, g naturally gives rise to an invariant volume form V_M \in \Omega^d(M). Let \Sigma be a smooth embedded submanifold of dimension d-1 in ... 1answer 79 views ### Exterior derivative commutes with postcomposition by symmetric multilinear functionals? Let \frak{g} be a finite-dimensional real Lie algebra, \varphi: \bigotimes^l \frak{g} \to \mathbb{R} a symmetric multilinear functional, and \psi \in \Omega^k(M; \bigotimes^l \frak{g}) a \... 2answers 136 views ### How do I derive this formula from gauge theory? This is Exercise 3.4.14 in R. W. Sharpe's Differential Geometry. Suppose G is a Lie group with Lie algebra \mathfrak{g} and H is a Lie subgroup of G. Let \theta be a \mathfrak{g}-... 1answer 62 views ### Problem proving Cartan's identity There is a famous identity stating that, if X is a field and \omega a form, then:$$\mathcal{L}_X\omega=\iota_Xd\omega+d\iota_X\omega.$$I'm trying to prove it. Thanks to Anthony Carapetis, I ... 1answer 43 views ### The formula for the differential of a vector-valued function If we have a vector, \,U=U\left(x_1,x_2,x_3\right), in the coordinate axis \left(x_1,x_2,x_3\right), then why does the following differential relation hold?$$ dU= \left(\frac{\partial U}{\...
I think I have a little confusion with index notation (concerning p-form). For example, do the coordinates on $\mathbb{R}^n$ $\{x^1,x^2, \dots , x^k\}$ and $\{x^{i_1},x^{i_2}, \dots , x^{i_k}\}$ mean ...