Questions tagged [differential-forms]

For questions about differential forms, a class of objects in differential geometry and multivariable calculus that can be integrated.

A degree $k$ differential form on a smooth manifold $M$ is a quantity that can be integrated on $k$-dimensional submanifolds of $M$.

Formally, a degree $k$ differential form is an element of $\Omega^k(M) = \Gamma(M, \bigwedge^kT^*M)$ which is the vector space of smooth sections of the vector bundle $\pi: \bigwedge^kT^*M \to M$; a section is a map $\alpha : M \to \bigwedge^kT^*M$ such that $\pi\circ\alpha = \operatorname{id}_M$. In particular, if $\alpha \in \Omega^k(M)$, for each $x \in M$, $\alpha(x) \in \bigwedge^kT^*_xM$; that is, $\alpha(x)$ is an alternating map $(T^*_xM)^k \to \mathbb{R}$.

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Justifying the "Physicist's method" for ODEs using differential forms

I need some help in untangling and solving the following exercise: Let the curve $c:[a,b] \to \mathbb{R}^2, t \mapsto (t, y(t))$ be a solution for the ODE $$ y'(x) = f(x, y(x)). $$ Justify the "Physicist's method" (no offense intended) of…
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Intuition behind an integral identity

A proof for the identity $$\int_{-\infty}^{\infty} f(x)\, dx=\int_{-\infty}^{\infty} f\left(x-\frac{1}{x}\right)\, dx,$$ has been asked before (for example, here), and one answer to that question actually generalized this identity to…
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Explain densities to me please!

When it comes to integration on manifolds, I speak two languages. The first is of course the language of differential forms, which is something I am relatively well acquinted with. The second language is what is often used in general relativity…
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Why do we care about differential forms? (Confused over construction)

So it's said that differential forms provide a coordinate free approach to multivariable calculus. Well, in short I just don't get this, despite reading from many sources. I shall explain how it all looks to me. Let's just stick to $\mathbb{R}^2$…
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Is there a Stokes theorem for covariant derivatives?

A $V$-valued differential $n$-form $\omega$ on a manifold $M$ is a section of the bundle $\Lambda^n (T^*M) \otimes V$. (That is, the restriction $\omega_p$ to any tangent space $T_p M$ for $p \in M$ is a completely antisymmetric map $\omega_p : T_p…
Turion
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How to calculate the gradient of $x^T A x$?

I am watching the following video lecture: https://www.youtube.com/watch?v=G_p4QJrjdOw In there, he talks about calculating gradient of $ x^{T}Ax $ and he does that using the concept of exterior derivative. The proof goes as follows: $ y =…
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Are Clifford algebras and differential forms equivalent frameworks for differential geometry?

I recently discovered Clifford's geometric algebra and its application to differential geometry. Some claim that this conceptual framework subsumes and generalizes the more traditional approach based on differential forms. Is this true? More…
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Where can I learn about complex differential forms?

So I'm a 3rd year grad student in number theory/modular forms/algebraic geometry, and I've worked with differential forms from an algebraic point of view without ever knowing what they really are. I'd like to fix this. I've recently read a section…
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What is the motivation for differential forms?

I am that point in my mathematical career where I am learning differential forms. I am reading from M.Spivak's Calculus on Manifolds. So far I have gone over the tensor and wedge products and their properties, defined forms, learned of their…
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Inducing orientations on boundary manifolds

Given a $k$-manifold $M$, such that $\partial M$ is a $(k-1)$-manifold, there is a standard way in which $\partial M$ inherits the orientation of $M$. So if $M$ is oriented by the form field $\omega$, then…
user21725
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Understanding proof of Cartan's magic formula: $L_X = i_X \circ d+d \circ i_X$

A possible proof of Cartan's magic formula $$L_X = i_X \circ d+d \circ i_X$$ is to follow the steps: Show that two derivations on $\Omega^{\bullet}(M)$ commuting with $d$ are equal iff they agree on $\Omega^0(M)$. Show that $L_X$ is a derivation on…
Seirios
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Proof of residue theorem (residue formula) for differential forms on curves over an arbitrary closed field.

I have been reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. They give a residue formula like this: Let $\mathbb{k}$ be an algebraically closed field and $X$ be a smooth projective curve over $\mathbb{k}$. Let…
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Geometric intuition behind pullback?

I am having hard time with forming a geometric intuition of pullback and pushforward. The definition the book gives is like this: There are two open sets, $A$ and $B$. There is a dual transformation of forms $\alpha^*$ between the forms on $A$ and…
Marion Crane
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Understanding the definition of interior product of differential forms as $(\iota_X\beta)(Y)=\beta(X,Y)$

The interior product of a 2-form $\beta$ and vector field $X$ is defined by $(i_X\beta)(Y)=\beta(X,Y)$ where $Y$ is a vector field. This is the definition of a 2-form (and it's similar for a p-form, just extended) but I don't understand what this…
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Why are de Rham cohomology and Cech cohomology of the constant sheaf the same

I am comfortable with de Rham cohomology, sheaves, sheaf cohomology and Cech cohomology. I am looking to prove the following theorem: If $M$ is a smooth manifold of dimension $m$, then we have the following isomorphism for each $k \leq m$ …