# Tagged Questions

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

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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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### Direct proof that the wedge product preserves integral cohomology classes?

Let $H^k(M,\mathbb R)$ be the De Rham cohomology of a manifold $M$. There is a canonical map $H^k(M;\mathbb Z) \to H^k(M;\mathbb R)$ from the integral cohomology to the cohomology with coefficients ...
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### Homework: closed 1-forms on $S^2$ are exact.

From the 2008 UCLA Geometry-Topology qualifying exam: let $\theta$ be a $1$-form on $S^2$ with $d \theta = 0$. Construct a function $f$ on $S^2$ with $d f = \theta$. I'm not very confident in my ...
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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 ...
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### Writing Integrals using Differential Forms

Consider some smooth curve $C \subset \mathbb{R^n}$ and $\gamma:[a,b] \subset\mathbb{R}\rightarrow C$ a parametrisation of $C$ and a continuous vector field $K:\mathbb{R^n} \rightarrow \mathbb{R^n}$. ...
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### Integral of wedge product of two one forms on a Riemann surface

I'm having trouble verifying an elementary assertion made in this answer on MathOverflow. It seems more like a math.stackexchange question, so I'm asking it here. Anyway, the assertion is as follows ...
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### Good intro to differential forms

I am looking for an intro book to learn about diff forms, maybe undergrad. Reading sentences like "Let M be a smooth manifold. A differential form of degree k is a smooth section of the kth exterior ...
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### Interior product of differential forms

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,...
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### Why are differential forms more important than symmetric tensors?

In differential geometry, differential forms are totally anti-symmetric tensors and play an important role. I am led to wonder why do we not study totally symmetric tensors as much as forms. What ...
This question can look like a duplicate of this one, but it's kind of different. I'm trying to relate some geometrical meanings I've seem in some books to the definition of differential forms in $\... 1answer 647 views ### Information captured by differential forms My advanced calculus class is currently doing differential forms and I have a hard time really understanding what they are all about. I can read the proofs of the theorems given in Rudin's PMA chapter ... 1answer 144 views ### What makes differential forms special There are two natural structures defined on differential manifolds, namely tangent bundle and its dual cotangent bundle. An element of tangent bundle$TM$at a base point$p\in M$can be described by ... 1answer 1k views ### Closed not exact form on$\mathbb{R}^n\setminus\{0\}$I'd like to construct a closed but not exact$n-1$-form$\omega$on$\mathbb{R}^n\setminus\{0\}$in analogy to the winding form: $$\frac{x~dy-y~dx}{x^2+y^2}$$ I think something like $$\omega=\frac{\... 3answers 214 views ### What is the relation between dx in elementary calculus and dx in differential geometry? I've recently started studying differential geometry and was really hoping that in doing so I'd finally have an answer to something that's been bugging me since I first learnt calculus - what is dx?!... 1answer 377 views ### Maurer-Cartan 1-form Can anyone help me with the following? Let \rho be the right-invariant Maurer-Cartan 1-form$$\rho = dg\ g^{-1}$$I want to show that the MC equation$$d\rho - \rho \wedge\rho = 0$$holds. So ... 0answers 185 views ### algebraic$1$-forms vs analytic$1$-forms First let's fix some definitions: Definitions: Complex manifold (of dimension n): Is a locally ringed space$(X,\mathscr F)$, where there is an open cover$\bigcup_{i\in I} U_i=X$such that$(U_i,\...
Let $\alpha$ be a closed $3$-form on $\mathbb{R}^{4} \setminus \{ 0 \}$. Let $i: S^{3} \hookrightarrow \mathbb{R}^{4}$ be the canonical embedding of $S^{3}$, and suppose that \$ \Omega := {i^{...