# Tagged Questions

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

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### How to calculate gradient of $x^TAx$

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 ...
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### Differential Forms and Vector Fields correspondence

Barrett O'Neill's Differential Geometry book says that Classical vector analysis avoids the use of differential forms on $\mathbb{R}^3$ by converting 1-forms and 2-forms into vector fields via the ...
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### Is there a nowhere $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a section of $\xi$?

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. Is there a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector field $X$ which is a ...
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### When does a null integral implies that a form is exact?

It is trivial to prove that the integration of a $(n-1)$ exact form on the boundary of a $n$-manifold is 0. What about the contraposative ? If the integration of a $(n-1)$-form on the boundary of a $n$...
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### What does a standalone $dx$ mean?

Some literature uses $dx$, in the context of differential equations, in a confusing way without defining what it really stands for: $Mdx + Ndy = 0$ Does it mean one of the following or something ...
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### Independence of $H(f)=\int_M \alpha \wedge f^* \beta$ on choice of $d\alpha=f^*\beta$?

I came across the following UCLA qual question while studying for my upcoming qual: Let $f: M^{4n-1} \to N^{2n}$ be a smooth map between closed connected oriented manifolds of the indicated ...
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### Pullback distributes over wedge product

I'm looking to prove that the pullback of a smooth function distributes over wedge product, i.e. $$\varphi^*(\omega \wedge \eta) = \varphi^* \omega \wedge \varphi^* \eta.$$ Here the question is ...
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### Manifold is not orientable

Let $M$ be a manifold of dimension $n$ such that there exist two charts $(U_a,\phi_a)$ and $(U_b,\phi_b)$ such that $U_a,U_b$ are connected and $U_a\cap U_b\ne\emptyset$. Moreover the transition ...
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### Compute $\int_cd\omega$ and $\int_{\partial c}\omega$

Question: Let $c:I^2\rightarrow\mathbb{R}^3$ be the singular $2$-cube given by $$c(s,t)=\left(\frac{1}{2}s^2,st,\frac{1}{2}t^2\right)$$Let $x=(x,y,z)$ denote the cartesian coordinates on $\mathbb{R}^3$...
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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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### 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 ...
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 $K:\mathbb{R^n} \rightarrow \mathbb{R^n}$. ...