# Tagged Questions

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

5k views

### 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 ...
466 views

689 views

### Why is arc length not a differential form?

I read that the arc length is not a differential form. But I don't understand why it isn't. I understand that differential forms are integrands and arc length is an expression which is integrable. ...
1k views

### 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 ...
62 views

### 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 ...
1k views

### 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 $n$...
151 views

### 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 ...
267 views

1k views

### 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 ...
307 views

### What is the intuition behind the definition of the differential of a function?

What is the intuition behind the definition of a differential of a function in differential geometry? i.e. $$df(p)(v_{p}) =v_{p} (f)(p)$$ where $v_{p} \in T_{p} M$ is a vector in the tangent space to ...
1k views

### Intuition behind $dx \wedge dy=-dy \wedge dx$

I was re-reading this old book of mine; and I noticed that in defining the rules of differential forms, it "makes sense" that we have the rule $dx \wedge dx=0$ because if $dx$ is infinitesimal, then ...
710 views

### $\Omega=x\;dy \wedge dz+y\;dz\wedge dx+z\;dx \wedge dy$ is never zero when restricted to $\mathbb{S^2}$

We define a 2-form $\Omega$ on $\mathbb{R^3}$ by $\Omega=x\;dy \wedge dz+y\;dz\wedge dx+z\;dx \wedge dy$. How can I show that $\Omega|_\mathbb{S^2}$ is nowhere zero? Before proving that how can I ...
991 views

### Integration of a differential form along a curve

Given the differential form $\alpha = x dy - \frac{1}{2}(s^2+y^2) dt$, I'd like to evaluate $\int_\gamma \alpha$ where $\gamma(s)=(\cos s,\sin s, s)$ and $0\leq s\leq \frac{\pi}{4}$. When attempting ...
176 views

### show that $\omega$ is exact if and only if the integral of $\omega$ over every p-cycle is 0

Let $M$ be an oriented smooth manifold and $\omega$ a closed $p$-form on $M$. Show that $\omega$ is exact if and only if the integral of $\omega$ over every $p$-cycle is $0$. In particular, how to ...
116 views

The motivation usually given to differential forms is that they generalize vector calculus nicely. That's true, but there are also $k$-vectors, i.e., objects from $\Lambda^k(V)$ instead of $\Lambda^k(... 2answers 371 views ### checking if a 2-form is exact Consider the 2-form $$\sigma=\frac{x_1 dx_2 \wedge dx_3 + x_2dx_3\wedge dx_1+ x_3 dx_1 \wedge dx_2}{(x_1^2+x_2^2+x_3^2)^{3/2}}.$$ I need to show if it is exact or not. Suppose it is exact, then there ... 1answer 661 views ### 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 ... 1answer 300 views ### When can a functional be written as the integral of a 1-form? Let a real, smooth manifold$M$be given. Let$\Gamma$denote the set of all path segments on$M$, namely the set of all paths of the form$\gamma:[a,b]\to M$. Let$Q:\Gamma\to\mathbb R$be a ... 1answer 334 views ### Is$ds$a differential form? I am somewhat confused as to whether$ds$(line element) is actually a differential form... we have (in$\mathbb{R}^2$): $$ds^2 = dx^2 + dy^2$$ Differential 1-forms are supposed to be linear ... 2answers 204 views ### Interpretation of$p$-forms Let$M$be a smooth manifold, let$C^{\infty}(M)$be set of all smooth functions from$M$to$\mathbb R$and let$Vec(M)$denote the set of all vector fields on$M$. A$1$-form on$M$is a$C^{\infty}(...
Let $\omega$ be a $2$-form on $\mathbb{R}^3\setminus\{0\}$ defined by $$\omega = \frac{x\,dy\wedge dz+y\,dz\wedge dx +z\,dx\wedge > dy}{(x^2+y^2+z^2)^{\frac{3}{2}}}$$ Show that $\omega$ is ...