2
votes
1answer
70 views

How to define integration over the boundary of a curve?

When learning about Stokes' theorem ($\int_{\partial \Omega} \omega=\int_{\Omega} \mathrm d \omega$), we are told that it is just a generalization of the 2nd Fundamental Theorem of Calculus $(\int_a^b ...
5
votes
1answer
49 views

Evaluating differential forms.

Can someone please check my work? It's an exercise from Barret O'Neill's Elementary Differential Geometry. I want to be really sure that my understanding of this is right. I see that the forms ...
2
votes
0answers
54 views

An error applying the Stokes theorem?

M is the surface $z=x^2+y^2$ with standard orientation for $x^2+y^2\leq 1$ and $\varphi = 4x^2ydy+z^2dz$ I'd like to verify that $\int_Md\varphi=\int_{\partial M}\varphi$, which I did, but ...
1
vote
2answers
41 views

Integrating a differential form inside a cylinder

Let S be cylinder given by $x^2+y^2=1$ between $z=1$ and $z=3.$ For $\varphi=e^xdx\wedge dy+ ydz\wedge dx+xdy\wedge dz$ find $\int_S\varphi$. I managed to finish the problem, but I'm getting ...
2
votes
1answer
45 views

Flux Integral - where did I go wrong?

S is the graph $z=25-(x^2+y^2)$ over the disk $x^2+y^2\leq 9$ and $\varphi = z^2dx\wedge dy$. Find $\int_S \varphi$. According to the book the answer is $3843\pi$, but the answer I got is ...
1
vote
2answers
62 views

The property of an integral manifold of differential form

Let's define the $1$-form $\eta$ on $\mathbb{R}^3$ by formula: $$\eta = A(x,y,z)\;\mathrm{d}x + B(x,y,z)\;\mathrm{d}y + \mathrm{d}z \text{.}$$ And let's assume that $$\mathrm{d}\eta \wedge \eta = ...
1
vote
0answers
32 views

Stoke's theorem explanation

Firstly, Wikipedia informs me that Stoke's theorem might mean two different things so I'm pretty sure I'm talking about the general one. I'll just state what I understand from it and I'd like someone ...
2
votes
0answers
26 views

Prove Green's theorem for circles

So the problem is in the title. The rules are that I can't split the circle into "rectangles" and I can't use pull-back. I tried to do something similar to the proof on unit squares. The problem is ...
1
vote
1answer
35 views

Integrate the differential form over a cardioid

$\omega=\dfrac{-ydx+(x-1)dy}{(x-1)^2+y^2}$ Calculate $\int_C\omega$ where $C...r=1+\cos\varphi$ (positively oriented) I'm still pretty lost when it comes to differential forms but as far as I ...
0
votes
0answers
20 views

Integrating a differential form over two oriented line segments

The problem is the following: Integrate the differential form $(\cos x \arctan e^x-y)dx+(2xy-y^2)dy$ over two oriented line segments $AB$ and $BC$ where $A=(0, -1), B=(1, 0)$ and $C=(0,1)$ I'm ...
3
votes
2answers
53 views

Using Stokes's theorem to calculate a value of integral

Use Stokes's theorem to calculate the integral $$I= \int_\Gamma (x^2+2y)dx+(y+z)dy+(z^2+x^2)dz$$ where $\Gamma$ is the boundary of $$\gamma=\left\{ (x,y,z):3x+y+3z=3,x\ge0,y\ge0,z\ge0\right\} $$ ...
0
votes
1answer
26 views

Proving the pseudosphere is regular and orientable.

The textbook I'm using define the tractrix by $ T=\{(\sin t, \cos t+\log (\tan (t/2))):0<t\leq\pi\}$ and define the pseudospher being the tractrix roting around the $z$-axis, I have to prove that ...
4
votes
1answer
75 views

Explanation of differential forms and notation

I'm doing multivariable calculus and I'd love if someone could shed some light on things that confuse me. When we did integrals of real functions with real variables, the $dx$ that was in every ...
1
vote
1answer
64 views

A connection to Stoke's Theorem (I think)

This is homework. I just finished a question regarding double integration over the unit sphere involving pullbacks of differential forms to provide context (course is advanced Calculus). The question ...
2
votes
4answers
90 views

Show that the form $w$ is closed but not exact

Let $w=\dfrac{-y}{x^2+y^2}dx+\dfrac{x}{x^2+y^2}dy$ Showing that $w$ is closed is easy. Just calculate $dw$ and you'll get 0. But how do I show that $w$ is not exact? In other words, I need to ...
1
vote
0answers
84 views

Line integral - should I parametrize the square?

I have the following $1-\text{form}$ defined: $$\omega = \displaystyle\frac{2xy}{(1-x^2)^2+y^2}\mathrm{dx}+\displaystyle\frac{1-x^2}{(1-x^2)^2+y^2}\mathrm{dy}$$ I'd like to find ...
6
votes
2answers
106 views

Relation between exterior (second) derivative $d^2=0$ and second derivative in multi-variable calculus.

What does an exterior (second) derivative such as in $d^2=0$ have to do with second derivatives as in single- or multi-variable calculus? Is this a correct start: Calculus derivatives are good for ...
10
votes
5answers
261 views

Is line element mathematically rigorous?

I know differentials (in a way of standard analysis) are not very rigorous in mathematics, there are a lot of amazing answers here on the topic. But what about line element? $$ds^2 = dx^2 + dy^2 ...
1
vote
1answer
47 views

Proving Borsuk-Ulam with Stokes

What is the easiest way to deduce the Borsuk-Ulam theorem in the case $n=2$ by using integration on manifolds and Stokes theorem? So I want to prove the following: Given a map $f\colon S^2\rightarrow ...
2
votes
1answer
58 views

Product integration of differential forms

Let $\alpha, \beta$ two forms continuous with compact supports and maximum degree on surfaces oriented $M, N $ respectively. consider $\pi_M:M\times N \rightarrow M$ and $\pi_N:M\times N \rightarrow ...
0
votes
0answers
65 views

differentiable equal r-forms

Let $\alpha$, $\beta$ be two $r$-forms continuous in $U\subseteq \mathbb{R}^n$ open. If $\int_M \alpha =\int_M \beta $ for all surface $M\subseteq U$   dimension $r$, compact, with boundary, then ...
2
votes
3answers
101 views

Simple criteria for “closed $\Longrightarrow$ exact”

In determining whether a closed form is an exact form, there is a lot of differential geometry definitions etc. that come in. I'm interested: what is the dummy, Calc III version of when closed implies ...
5
votes
1answer
143 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 ...
4
votes
5answers
786 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 ...
5
votes
2answers
265 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. ...
11
votes
3answers
472 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 ...
10
votes
1answer
357 views

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 $K:\mathbb{R^n} \rightarrow \mathbb{R^n}$. ...
3
votes
2answers
121 views

Prove that $\frac{1}{2\pi}\frac{xdy-ydx}{x^2+y^2}$ is closed

I would like to prove that $\alpha = \frac{1}{2\pi} \frac{xdy-ydx}{x^2+y^2}$ is a closed differential form on $\mathbb{R}^2-\{0\}$ . However when I apply the external derivative to this expression ...
1
vote
1answer
92 views

A differential form to compute the k-volume of a k-parallelogram in n dimensions

Computing the k-volume of a k-parallelogram (i.e. a parallelogram spanned by k n-dimensional vectors) in n dimensions is straightforward: Let $P=[\overrightarrow{v_1},...,\overrightarrow{v_k}]$, then ...
1
vote
1answer
62 views

$k$-forms on $\mathbb{R}^n$

Given an expression like $$ dx_1\wedge dx_2 \wedge dx_4 \left( \begin{bmatrix} 1\\2\\3\ \end{bmatrix} \ , \ \begin{bmatrix} 4\\5\\6 \end{bmatrix} \ , \ \begin{bmatrix} 7\\8\\9 \end{bmatrix} \right) \ ...
2
votes
3answers
73 views

Whats the connection between formss and vector fields?

I heard someone talking about how vector fields are the kernels of forms. Can someone give me a detailed explanation about how this works? Thanks.
1
vote
1answer
44 views

understanding simple multivariable integrals in terms of differential forms

I am learning a bit about differential forms: defining differential forms in terms of elementary forms, integrating forms over parametrized domains, etc. I would like to relate this to my previous ...
1
vote
1answer
158 views

How general formulation of Stoke's theorem relate to Kelvin-Stokes theorem

I asked a similar question, but I realized the question is too vague and it's better to start a new one: We know that there are two usually used formulations of Stoke's theorem. One is vector ...
1
vote
1answer
199 views

Example of differential form usage of Stoke's theorem

There are many examples that show how Kelvin-Stokes theorem is used. But I would like to see an example that uses differential form usage of Stoke's theorem and is hard or impossible to solve by ...
2
votes
1answer
145 views

Finding the winding number of a curve

Let $\gamma(t)=(r \cos t,r \sin t)$, for some $r>0$, and let $\Gamma$ be a $C^2$-curve in $\mathbb R^2-\{\bf 0\} $, with parameter interval $[0,2 \pi]$, with $\Gamma(0)=\Gamma(2 \pi)$, such that ...
2
votes
1answer
220 views

Sphere volume and generalized Stokes' theorem

The area of a circle is $\pi r^2$, and the circumference is the derivative of this: $2\pi r$. The same holds in one higher dimension: the volume of a sphere is $\frac{4}{3} \pi r ^3$ and the ...
2
votes
2answers
147 views

How is differential form different from ordinary calculus objects?

I am going to learn differential form soon, but after reading some introductory parts of my texts, I couldn't get why differential form is needed and how it is different from ordinary mathematics ...
4
votes
1answer
117 views

Integration over a surface

Let $S$ be given by $$S= \left[(x,y,z) \in \Bbb{R}\;|\; x^2+y^2+z^2+xy+xz+yz=\frac12 \right]$$ and $$\omega = xdy \wedge dz\, -\, \frac {2z}{y^3} \, dx\wedge dy \,+\, \frac1{y^2}dz\wedge dx $$ ...
4
votes
2answers
142 views

Function on $\mathbb{R}^{2}-\{0\}$.

Does there exist any compactly supported function $f= (f_1,f_2): \mathbb R^2-\{0\}\to \mathbb R^2$ such that $$\frac{\partial}{\partial x_2}f_1=\frac{\partial}{\partial x_1}f_2.$$ Also there does not ...
4
votes
2answers
231 views

algebraic manipulation of differential form

Suppose $\phi_1, \phi_2, \dots, \phi_k \in (\mathbb{R}^n)^*$, and $\mathbf{v}_1, \dots, \mathbf{v}_k \in \mathbb{R}^n$ $(\mathbb{R}^n)^*$ stands for the space of all linear transformations that goes ...
2
votes
0answers
64 views

How to apply Gauss's theorem when the metric is unknown

Let $f:U \to \mathbb{R}^3$ be a surface, where $U=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|<3, |u^2|<3\}.$ Consider the two closed square regions $F_1=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|\leq1, ...
2
votes
1answer
252 views

Taking the exterior derivative of a 0-form

I'm attempting to show that $dg(\vec{x})=\alpha$, where $\alpha=\Sigma^n_{i=1}f_idx_i$ and $g(\vec x)={1\over{p+1}}\Sigma_{i=1}^nx_if_i(\vec x)$...and $d$ is the exterior derivative. The $f_i$ are ...
1
vote
1answer
169 views

Computing a Differential Form

Apologies in advance, I don't know TeX, so this might look a bit gross... I'm given a 1-form $A=f_1dx_1+...+f_ndx_n$, infinitely differentiable and closed on $R^n$. I want to show that $dg=A$ for ...
1
vote
1answer
92 views

Show that $\omega = d(I\omega)$ if $d\omega = 0$

Let $\omega = P\ dx + Q\ dy$ be a 1-form on $\mathbb{R}^2$. Also, define a 0-form $I\omega({\bf x}) = I\omega(x, y)$ by $$ I\omega({\bf x}) = \int_0^1 P(t {\bf x}) x + Q(t {\bf x}) y\ dt.$$ I would ...
9
votes
1answer
469 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 ...
0
votes
1answer
308 views

Integrating factor and differential 1-forms

I am working on the following exercise: The function $f$ is called an integrating factor for the 1-form $\omega$ if $f({\bf x}) \neq 0$ for all $\bf x$ and $d(f\omega) = 0$. If the 1-form $\omega$ ...
3
votes
1answer
153 views

iterated integration and shuffle product

Recall that a $(r,s)$-shuffle $\sigma$ is a permutation of $r+s$ letters such that $\sigma^{-1}(1) < \cdots< \sigma^{-1}(r)$ and $\sigma^{-1}(r+1) < \cdots < \sigma^{-1}(r+s)$. If ...