# Tagged Questions

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

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### Integrating differential forms on curves [closed]

How can I integrate the differential form $$\omega=x\,dx+y\,dy+z\,dz$$ in $\mathbb R^3$ on the curve $$c:[0,2\pi]\to\mathbb R^3: t\mapsto (e^{t\sin t}, t^2-2\pi t, \cos \frac{t}{2})?$$ Some advice ...
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### Explicit isomorphism $H (E, E^0) \cong H_{cv} (E)$

Yo! I've been trying to understand better why $$H (D, D-\{x\}) \cong H_c (D)$$ for a disk $D$ and, more generally, why $$H (E, E^0) \cong H_{cv} (E)$$. In general, the first isomorphism can be seen as ...
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### What do mathematicians mean when they say “form”?

As in differential form, modular form, quadratic form? I'm sorry if this is a really silly question.
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### Finding a two-dimensional chain

Let $$T=\{(x,y,z,w,)\in R^4:x^2+y^2=z^2+w^2=\frac{1}{\sqrt 2}\}$$ and $$\omega=dx\land dy + dz\land dw$$ in $\mathbb R^4$. How do I find a two-dimensional chain $C$ where $T$ is its trace? And how can ...
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### Domain simply connected for a differential form $\omega$
I have the domain $\Omega=\{(x,y): 2y+x>0\}$ and the differential form $$\omega=\frac{y}{2y+x}dx+\left(\log (2y+x)+\frac{2y}{2y+x}\right)dy.$$ I would like to evaluate $\int_{\gamma} \omega$ ...
### Proving a formula for the exterior derivative of a specific $k$-form, given in base representation
Let $U \subseteq\mathbb{R}^n$ be open, and let $\omega$ be an $(n-1)$-form that's given by \omega = \sum_{i=1}^n (-1)^{i-1} F_i\, dx_1 \wedge\dots\wedge dx_{i-1} \wedge dx_{i+1} \wedge\dots\wedge ...