# Questions tagged [exterior-derivative]

For questions related to exterior derivative. The exterior derivative of a function $f$ is the one-form $df=\sum_i\frac{\partial f}{\partial x_i}dx_i$ written in a coordinate chart $(x_1,\dots,x_n)$.

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### Is the exterior derivative of a form with values on a vector space well-defined?

I've recently asked a question on Physics SE about how one should interpret a particular object in the local expression for the spin covariant derivative. Namely, the object was $\mathrm{d}\psi(X)$, ...
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### Infinitesimal Interpretation Of Exterior Derivative

I have a question regarding the conceptual understanding of the exterior derivative. I've read that one can view a $k$-form on an $n$-dimensional manifold as a collection of infinitesimal (oriented) ...
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### Is $\text{d}^{\dagger}\text{d}$ or $\text{d}\text{d}^{\dagger}$ by itself a valid operator?

If I simply consider just one of the combinations $\text{d}^{\dagger}\text{d}$ or $\text{d}\text{d}^{\dagger}$ both of them take from $\Omega^r(M)\to \Omega^r(M)$ for some manifold $M$. But do they ...
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### Exterior derivative of 1

In one of the examples of Loring W. Tu's Book "Introduction to Manifolds" (section 19.7 of 2nd ed.) the author takes the exterior derivative on both sides of the equation $x^2 + y^2 = 1$ ...
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### Prove the coordinate-free formula for the exterior derivative

I can prove the following : There exist a unique map $d:\Omega^k(M)\to \Omega^{k+1}(M)$ such that $d$ is $\mathbb{R}$-linear $d^2=0$ For a $0$-differential form (a function) $d$ reduces to the usual ...
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I am thinking about the 2 dimensional vector calculus operation "grad perp" $\nabla^\perp := (-\partial_y , \partial_x)$. According to these notes, this can be defined using the Hodge star ...
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### Difficulty understanding a step in Weintraub's Differential Forms Textbook:

In Weintraub's Differential Forms, he gives this property: Let $f$ be a differentiable function defined on a region R of $\mathbb{R}^3$. Let $f_*$ be the derivative of $f$ with respect to some ...
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### Request of help for a problem on 2-forms

Let $y = f(x)$ and $z = g(x, y)$ be real functions of one variable $x$ and two variables $(x,y)$ respectively. Suppose $$dz \wedge dx =0.$$ What conclusion can be drawn from this statements? My ...
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### Closed $n$ form from Harvard Qualifying Exam

In the most recent Harvard Qualifying exam, one is asked to prove that if $M$ is a compact oriented manifold, and there are two nonvanishing orientation forms $\xi$ and $\eta$ who's integrals over $M$ ...
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### Why can the exterior derivative be interpreted as in this picture?

This paper provides a geometric intuition of differential forms. On page $5$ it reads: "Consider the case $xdy$. The number of horizontal lines is roughly proportional to $y$. In other words the ...
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### Understanding Coulomb's law in the language of differential forms

I began studying differential forms to gain a deeper understanding of the Maxwell's equations. Let's say we have a $2$-form representing the electric flux density (the $D$-field). We can take the ...
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### Intuition for the commutation of the pullback and the exterior derivative

I know that there are already several questions to this topic but I haven‘t seen a satisfactory answer. Deriving that the pullback and the exterior derivative commute is no problem but I want to know ...
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### Lie derivative of a differential form

I have a differential $1$-form $\omega = x\mathrm{d}x + x\mathrm{d}y$ and I need to find its Lie derivative along $X = (x+y)\partial_{x} - 2y\partial_{y}$. The first approach is by using Cartan ...
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### What does C.H. Edwards mean by "orientation" in the development of Stokes's theorem in n dimensions?

Edit: I am inclined to believe that the overall answer is going to be: it doesn't matter which sense we call positively oriented as long as we are consistent during our traversal. The following is ...
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### Generalization of Helmholtz theorem to differential forms?

In vector calculus, Helmholtz theorem says the divergence and curl of some vector field uniquely determines the vector field itself (with appropriate boundary conditions). Can this be generalized to ...
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### Are $\phi$ and $\hat z$ 'dual' in $\mathbb R^3$?

Consider $\mathbb R^3$ and the following equation $$\star d\omega=df$$ where $f$ is a 0-form and $\omega$ is a 1-form. My question is whether this equation is satisfied by \...
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### What is the name for this relationship between a 1-form and a vector? [duplicate]

all. I have a question about Visual Differential Geometry and Forms - A Mathematical Drama in Five Acts (by Tristan Needham). This book shows two relationships between Forms and vector. And I have a ...
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### Pullback of an exterior differential system

An exterior differential system (EDS) is a pair $(M,\mathcal{E})$ consisting of a smooth manifold $M$ and a homogeneous differentially closed ideal $\mathcal{E}$ of the graded algebra $\Omega^*(M)$ of ...
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### Exterior derivative of a 2-form defined by composition of metric with an endomorphism

Im interested in calculating the exterior derivative of a 2-form defined by $\omega(x,y) = g(x,Ay)$ where $A \in \Gamma(End(T) )$ is skew and $g$ is some metric. I hope to reach some coordinate-...
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