Questions related to understanding line integrals, vector fields, surface integrals, the theorems of Gauss, Green and Stokes. Some related tags are (multivariable-calculus) and (differential-geometry).

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Defining divergence of vector field

Curl is defined (in the plane) by imagining a wheel of radius $\epsilon$ placed in $(a,b)$. Denote the region enclosed by the boundary of the wheel with $D_\epsilon$. Let's suppose our vector field ...
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3answers
59 views

Find dimension of the intriguing vector space

We are given vector space of polynomials over $\mathbb R$ of two variables with powers not higher than 2013. Let's consider subspace $V$ which contains such polynomials $f$, so following holds for ...
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2answers
38 views

Gauss's theorem in 2d: how can it be expressed in differential forms?

How do we express the 2d version of Gauss's theorem in the language of differential forms? In 3d, I know it is $$d \left(Fdydz + Gdzdx + Hdxdy\right) = F_x + G_y + H_z dxdydz$$ so by Stokes' ...
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42 views

derivation of divergence from nabla operator

For a two dimensional orthogonal curvilinear coordinate system $(t_1, t_2)$, we have the position vector $r$, where $h_i = | \frac{\partial r}{\partial t_i} |$ are the scale factors and $a_i$ are the ...
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1answer
11 views

Find the surface integral of some ellipsoid?

I got Stokes theorem all warmed up for this one! $$\int_{S}\int(Curl(\vec{F}))d\vec{s}$$ (That means delta cross F or curl of F) Where S is the ellipsoid $x^2 + y^2 + 2z^2 = 16$ And $\vec{F} = ...
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1answer
18 views

Find the area bounded by the hypercycloid

Parametrization : $x = acos^3(t), y = asin^3(t)$ $a>0$ If you can solve it for me that would be awesome.:D If not, can you give me some hints? Tell me how to set it up and stuff. It's solveable ...
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31 views

Is it correct that $\frac{d\theta}{d\varphi'}=\left(\frac{d\varphi'}{d\theta}\right)^{-1}$?

Let $\theta$ be $k\times1$ and $\varphi$ be $k\times1$. Then $\frac{d\theta}{d\varphi'}$ is a $k\times k$ matrix with entries $\frac{d\theta_{i}}{d\varphi_{j}},\,\, i,j=1,...,k$. I was wondering if ...
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48 views

understanding Green's theorem Intuition

The idea of it is to find the area of a region, yet I keep seeing vector fields popping up all over the place. Take this example from my text book: Find the region enclosed by the two graphs: $y = ...
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2answers
89 views

Why does Stokes theorem apply to this situation?

I'm thinking Green's theorem or stokes theorem, but I don't know. It has been driving me crazy all day. Help me out here! And if you don't want to help because you know it's homework, give me some ...
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1answer
34 views

Find the area bounded by the hypocycloid?

I have the answer. The hypobloid has parametrization = $x = acos^3(t)$ $y = asin^3(t)$ The explanation is you take a vector field $F(x,y) = (0, x) which has curl 1 than it says the area is equal to: ...
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41 views

Can this line integral problem be solved with Stokes theorem?

I have a feeling it could, or with some other theorem. $F(x,y,z) = (2xyz + \sin x)i + (x^2z)j + (x^2y)k$ $$\int_{c} F.ds$$ where $c(t) = (\cos^5(t),\sin^3(t),t^4)$ I tried it in differential form ...
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39 views

Is there a way besides integration by parts to solve this integral?

$$\int_{0}^{2\pi} -10\cos^9(t)\sin^4(t)t^4\,dt$$ Maybe a formula for this form or something?
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1answer
32 views

Evaluating a line integral through a vector field in 3 dimensions.

Let $\mathbf{F}(x,y,z) = (2xyz + \sin x)\mathbf{i} + (x^2 z)\mathbf{j} + (x^2 y)\mathbf{k}$. Evaluate the integral of $\mathbf{F}$ along $c$, where $c(t) = (cos^5(t), sin^3(t), t^4)$, $t \in [0, ...
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1answer
52 views

Is there a vector field that is the complete opposite of a conservative one

Is there a three-dimensional vector field such that for every non-selfintersecting closed curve (that is not just one point, to avoid degenerate cases) the respective line-integral on the curve ...
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1answer
73 views

Line integral + Work

$F=(z-y)i+(x-z)j+(2y-x)k$ Let $C$ be a curve formed by an intersection of the plane $2x-z=0$ with the cylinder of elliptical cross section $x^2+(y^2)/9=1$, assuming $y$ is parametrized along $C$ via ...
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1answer
53 views

Prove: If $\int_{\phi}\omega = \int_{\psi}\omega$ whenever $\phi$ and $\psi$ have the same endpoints, then $\omega=df$

This is an exercise that I have been assigned for homework. I don't really know how to approach it though. I know that $\int_{\phi}\omega$ only depends on the endpoints $\phi(a)$ and $\phi(b)$ where ...
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2answers
40 views

Clarification: What does it mean when “$\phi$ and $\psi$ are two smooth curves in $U$ with the same beginning and end points”

"$\phi$ and $\psi$ are two smooth curves in $U$ with the same beginning and end points" Does this mean: (A) $\phi:[a,b]\to U$ and $\psi:[a,b]\to U$ (B) $\phi:[a,b]\to U$ and $\psi:[c,d]\to U$ s.t. ...
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1answer
41 views

integrals of vector fields that yield vectors, not scalars

When I tried to think of how I'd answer this question, I realized that never in my undergraduate curriculum was I asked to compute the surface or line integral of a vector field. I don't mean I've ...
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1answer
64 views

Integral/Vector calculus $\oint_{\partial S} u \vec \nabla v \cdot d \vec \lambda=\int_S (\vec \nabla u)\times (\vec \nabla v)\cdot d\vec S.$

I am trying to show that $$ \oint_{\partial S} u \vec \nabla v \cdot d \vec \lambda=\int_S (\vec \nabla u)\times (\vec \nabla v)\cdot d\vec S $$ using Levi Cevita notation methods only. The Levi ...
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1answer
21 views

Vectors and Forces

A box weighting 294N is sitting on a ramp. If the ramp is inclined at an angle of 25 degrees to the horizontal, and there is a 40N force of friction, calculate the amount of force that must be ...
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1answer
16 views

$w$ is a form, find $\alpha$ s.t $d \alpha=w$

$w$ is a form, find $\alpha$ s.t $d \alpha=w$ $w=(2y-4)dy \wedge dz+(y^2-2x)dz \wedge dx+(3-x-2yz)dx \wedge dy$ $w$ is a 2-form and $d \alpha=w$ so $\alpha$ is a 1-form s.t: $\alpha =Mdx+Ndy+Pdz$ ...
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1answer
27 views

Let $\alpha= f\,dx_1 \wedge \cdots\wedge dx_n$; where $f$ is continuous on $A$. Show that $\int_ \Phi \alpha =\int_ \Phi f$

Let $A \subset \mathbb{R}_k$ be a rectangle (or box), and let $\Phi:A\to\mathbb{R}_k$, be the identity mapping. Let $\alpha= f \, dx_1 \wedge \cdots \wedge dx_n$; where $f$ is continuous on $A$. ...
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1answer
36 views

Let $d(fw)=0$ for $f\neq 0$ show that $w \wedge dw =0$ where w is a 1-form.

Let $d(fw)=0$ for $f\neq 0$ show that $w \wedge dw =0$ where w is a 1-form and f is a 0-form My attempt: Let $c=1/f$ then $cw \wedge d(fw)=0$ implies $cw \wedge (df \wedge w+f \wedge dw)=0$ implies ...
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15 views

If $w= \sum_{i<j}a_{ij} dx_i \wedge dx_j$ find $dw$

If $w= \sum_{i<j}a_{ij} dx_i \wedge dx_j$ show that $dw=\sum_{i<j<k}(\dfrac{\partial a_{ij}}{\partial x_k}-\dfrac{\partial a_{ik}}{\partial x_j}+\dfrac{\partial a_{jk}}{\partial x_i})dx_i ...
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21 views

Prove that if $\bigtriangledown \times F=0,$ then $\exists u$ such that $F= \bigtriangledown u$

If $\bigtriangledown \times F=0,$ then $\exists $ a function $u$ such that $F= \bigtriangledown u$ The converse is straightforward to prove, but this statement is giving a hard time. Any ...
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1answer
28 views

Line intergral around a closed path

Q: Evaluate the closed line intergral $ \oint xdy $ anti-clockwise around the triangle with vetricies $(a,0), (0,0),$ and $(0,b)$ For this section I've reduced the line sections to: $ C1: x = x, y = ...
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4answers
71 views

Proof: Force always perpendicular and motion in a plane implies that the trajectory is a circle

I am looking for a proof for a physics problem. Consider a particle which is subject to a force $\vec{F}(t)$ with $|\vec{F}(t)| = \text{const}$ which is always perpendicular to the velocity ...
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1answer
47 views

Line Integral Help (Vector Calculus)

I'm currently revising for a maths module that I am taking as part of my physics degree. I'm taking the exam tomorrow and I'm feeling pretty confident although upon attempting this line integral I ...
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1answer
45 views

Line Integrals in $\mathbb{R}^n$ and Differential 1-Forms

I am really lost in the notation of my book and am looking for some conceptual help. So... for a differential 1-form $\omega$, every value of $x\in \Omega$ maps to a linear function $\omega _x$, ...
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39 views

Field of vector fields

For every point $A$ outside a sphere with radius $a$, there's a field $$F= \frac{K}{r^4d^2} $$ where $r$ is distance between point $A$ and the center of the sphere, and $d$ is distance between point ...
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2answers
75 views

Intersection curve between a circle and a plane - Stokes theorem

What is the intersection curve between the circle $$x^2+y^2=1$$ and the plane $$x+y+z=0$$ If i am not wrong, I should solve the equation system \begin{align} x^2+y^2-1=0 \\ x+y+z=0 \end{align} But I ...
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33 views

Interior Products

Over on the Wiki page for interior products: http://en.wikipedia.org/wiki/Interior_product There is a line that says $\iota_X \alpha = \alpha(X) = \langle \alpha,X \rangle$ where $\alpha$ is a ...
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1answer
27 views

evaluating a flux integral

Question: "Region V, of unit volume, is bounded by the closed surface S. Given the vector field $\mathbf{F}=\langle 7x,2y,5z\rangle$, evaluate: $$\int_S \mathbf{F}\cdot\mathbf{dS}$$ I guessed that ...
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34 views

Dot Product and vector length

Hi! I am working on some online homework for my calc2 class that covers the dot product and I am really struggling with this one question. I understood how to solve part a, because we covered that ...
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1answer
59 views

Do line integrals of non smooth curves exist?

Wolfram says that the theorem of conservative fields is : The following conditions are equivalent for a conservative vector field on a particular domain $D$: For any oriented simple closed ...
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11 views

Question about vector field and field line

If I want to draw the lines of force to a vector field F and consider whether any results are physically reasonable, then i can start with to solve the differentiable equation $$ y'(x)= {F_y(x,y) ...
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0answers
24 views

Counting the roots of nonlinear systems of equations

I have a "nice" function (vector field) $$\mathbf{f}: \mathbb{R}^n \rightarrow \mathbb{R}^n$$ and I need to find how many roots (zeros) it has in a certain domain (hopefully prove that it has at most ...
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Integrate a divergence-free vector field

Suppose we are given a vector field $\overrightarrow{B}$ in $\mathbb{R}^3$ whose divergence is zero : $div(\overrightarrow{B})=0$. We want to find $\overrightarrow{A}$ such that ...
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56 views

Curl and unit vectors after a change of coordinates

Let $x,y,z$ be a system of coordinates with unit vectors respectively $\mathbf{u}_x$, $\mathbf{u}_y$, $\mathbf{u}_z$. Moreover, we have $\nabla = \displaystyle \frac{\partial}{\partial x} ...
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2answers
62 views

Give an informal reason why this cannot be the gradient of a functoin

Explain why $F(x,y) = \Big(\frac{-y}{x^2 + y^2}, \frac{x}{x^2+y^2}\Big)$ cannot be the gradient of a function (defined away from the origin). Can it be the gradient if we only require F and $f$ to be ...
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0answers
50 views

Green Identities via Differential Forms

What is the actual meaning of the Green identities: Is there a picture/geometric interpretation of these, as well as intuition going beyond the usual integration-by-parts meaning associated to ...
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2answers
25 views

Matrix and vector multiplication order

Assume $u\in \mathbb{R}^{m\times1}, X\in\mathbb{R}^{m\times m}, v\in\mathbb{R}^{n\times 1}, w\in\mathbb{R}^{n\times 1}$ and $m\neq n$. Then are the expressions $u^TX\,u\in \mathbb{R}$ and $v \cdot w ...
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2answers
72 views

True or False Stokes'/Gauss Theorem Problems

I'm having difficulties deciding the truths of the two following statements. The first one I believe is false, but I'm not entirely sure and the second one I don't know how to make heads or tales of. ...
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1answer
41 views

Differential Forms / Stokes' Theorem Problem

Problem: Let $w = (x + y)dz + (y + z)dx + (x + z)dy$ and let $S$ be the upper part of the unit sphere; that is, $S$ is the set of $(x,y,z)$ with $x^2+y^2+z^2 =1$ and $z\ge0$. $\delta$$S$ is the unit ...
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Elementwise normal to vector of unknowns and non-defined matrix multiplications

I wonder about statement (1). It is given that u is a column vector, A and B are constant, symmetric, square matrices of such size that the expressions on the left hand side of (1) are well defined. ...
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14 views

Differentiation of vector-function

Let $f(x) = e^{-x^Tx},$ where $x \in \mathbb{R}^n$. What will be the second derivative? The first is $~f'(x) = 2x^T e^{-x^Tx}$, and when I try to find the second, I confuse. It will be $$f''(x) = ...
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42 views

Question about vector field

If I want to draw the lines of force to a vector field F and consider whether any results are physically reasonable, then i can start with to solve the differentiable equation $$ y'(x)= {F_y(x,y) ...
1
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0answers
30 views

Covariant Derivative to study Holonomy

Can someone provide an explicit definition to the $\mathbb{R}^{3}$-covariant derivative? For instance, I don't understand the following calculation. Let $$E_1 = (-\cos(u) \sin(v), - \sin(u) \sin(v), ...
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1answer
27 views

Line integrals and vector fields

We had this example in class the other day, and the professor didn't not walk through how he obtained it. Compute $\int_C \vec f \cdot d\vec r = \langle 4x^3y^2 - 2xy^3, 2x^4y - 3x^2y^2 + ...
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2answers
103 views

Why is $\operatorname{Div}\big(\operatorname{Curl} F\big) = 0$? Intuition?

Why is $\operatorname{Div}\big(\operatorname{Curl} F\big) = 0$? Is there an intuitive explanation to what this means as well as an algebraic proof? Also I understand that $\operatorname{Curl} ...