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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5
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4answers
60 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 ...
1
vote
1answer
46 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 ...
1
vote
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 $\omega _x$, ...
1
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0answers
36 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 ...
1
vote
2answers
66 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 ...
1
vote
0answers
24 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 ...
0
votes
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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votes
3answers
32 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 ...
0
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1answer
42 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 ...
0
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0answers
10 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) ...
0
votes
0answers
22 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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0answers
22 views

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 ...
0
votes
1answer
51 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} ...
0
votes
2answers
60 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 ...
2
votes
0answers
45 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 ...
0
votes
2answers
24 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 ...
0
votes
2answers
62 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. ...
0
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1answer
38 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 ...
1
vote
0answers
27 views

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. ...
0
votes
0answers
12 views

Bubble inside field of charge

There is a charge distribution $\rho(\mathbf x)$ with electrostatic potential $\phi(\mathbf x)$ s.t for $|\mathbf x|<a,$ $\phi=0$ and for $|\mathbf x|=a$, $\phi(\mathbf x)=\Phi$. Show that $\Phi ...
1
vote
0answers
13 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) = ...
1
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0answers
40 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
vote
0answers
29 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), ...
0
votes
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 + ...
2
votes
2answers
96 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} ...
0
votes
2answers
30 views

Green's Theorem and Divergence (2D)

I am reading the book Numerical Solution of Partial Differential Equations by the Finite Element Method by Claes Johnson. In Chapter 1 he talks about the Possion Equation, and to prove that FEM ...
0
votes
1answer
28 views

Curl of a function with only angular dependence

Let a function in spherical coordinates $$\vec F(\vec r) = \int{ d^3\vec r\,' \vec j(\vec r\,') } \,e^{-ik\hat r \cdot \vec r \,'}$$ Where $\vec j$ is a vector function. So $\vec F$ only depends on ...
1
vote
0answers
25 views

Stoke's Theorem for an open cylinder

How do you use Stokes' Theorem to calculate the surface integral over a cylinder of $\nabla \times F$? Do you have to calculate the line integrals along the top and the bottom? If so, is this example ...
2
votes
0answers
60 views

Green's Theorem

Hey guys I am having difficulties in problem 5. I thought I understood it, but I suppose I was mistaken. I will now explain what I planned to do to solve this problem and where I got stuck. So I ...
0
votes
0answers
36 views

approximating a sphere

Suppose that $R$ is a simple connected region in $\mathbb{R}^3$, enclosing a volume $V$. I am looking at ways to approximate $V$ using spheroidal volume elements. The traditional approach is to use ...
2
votes
1answer
29 views

Surface integral is $0?$

I have a quick question: if $\mathbf{f}:\;\mathbb{R}^3\to\mathbb{R}^3$ is odd, in the sense that $\mathbf{f}(-\mathbf{v})=-\mathbf{f}(\mathbf{v})$ for any $\mathbf{v}\in\mathbb{R}^3$, and $S$ is a ...
0
votes
1answer
44 views

Stokes' Theorem and Surface Independence Failure

As we know, if $\vec{F}=\nabla\times\vec{A}$ then from Stokes' Theorem, $\iint_{S_1} \vec{F}\dot \,d\vec{S}=\iint_{S_2}\vec{F}\dot \,d\vec{S}$ where $S_1$ and $S_2$ have the same boundary. Does ...
0
votes
2answers
62 views

why are conservative vector fields curl-free?

The book told me that, if a vector field $\vec{F} = Mi + Nj$ is conservative, then $$ M_y = N_x $$ But why is this true?
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0answers
33 views

Three linearly independent vector fields

How can one find three linearly independent vector fields on $S^1\times S^2$? I know that $S^1\times S^2 \cong SO_3( \mathbb{R})$, i.e. the set of orthogonal $3 \times 3$ matrices with determinant ...
0
votes
0answers
26 views

vector field problems

I'm trying to review some problems on vector fields for the final, and would appreciate if someone can tell me whether my answers are right, so I know if I'm doing it correctly: $f$ is a vector ...
3
votes
1answer
35 views

Why should we expect the divergence operator to be invariant under transformations?

A lot of the time with vector calculus identities, something that seems magical at first ends up having a nice and unique proof. For the divergence operator, one can prove that it's invariant under a ...
0
votes
2answers
70 views

Find the flux of the vector field across the boundary of the cube

Find the flux of the vector $F=e^{xy} \hat{i} +e^{yz} \hat{j} +z \hat{k}$ across the boundary of $[0,1] \times [0,1] \times [0,1]$. Can someone tell me the setup of this problem?
1
vote
0answers
51 views

packing spheroids/ellipsoids

I am thinking about a problem I'm trying to formulate, from a mathematical analysis viewpoint. Suppose that you have a finite region in 3-space, and you want to populate it with spheroids or even ...
0
votes
0answers
69 views

How to convert vector field from cartesian to spherical

I have a vector field $A ( r) = \omega \times r$, where $r=(x,y,z)^T$ and now I want to express this field in cylindrical coordinates. How do I do this?
0
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0answers
45 views

How to find $\nabla$ in spherical coordinates

I want to derive(!) just a few components like the $\hat{e}_r$ component of the divergence operator in spherical coordinates and the $\hat{e}_{\phi}$ component of the curl operator in spherical ...
0
votes
1answer
31 views

Showing orthogonality of coordinate surfaces are orthogonal for oblate spheroidal co-ordinates.

So oblate spheroidal co-ordinates are defined as: $$x = \cosh R \cosθ \cos φ$$ $$y = \cosh R \cosθ \sin φ$$ $$z = \sinh R \sin θ .$$ To show the coordinate surfaces for $R$, $\theta,\phi$ are ...
0
votes
0answers
18 views

Constructing a vector field with a given divergence

Let $f$ be a scalar-valued trivariate function defined on some bounded domain, say $[a_1,b_1]\times[a_2,b_2]\times[a_3,b_3]$. I would like to construct a vector field ...
0
votes
1answer
28 views

Vectors and Planes

Let there be 2 planes: $x-y+z=2, 2x-y-z=1$ Find the equation of the line of the intersection of the two planes, as well as that of another plane which goes through that line. Attempt to solve: the ...
0
votes
0answers
26 views

Vector Question Help

A plane is determined by $(x,y,z) = (1,-1,0) + t(1,-1,2)$ and point $p(1,2,3)$. find point of intersection of $(1,4,-1)+s(-6,2,-4)$ with this plane. I tried this: given the data plane equation is:$$ ...
1
vote
1answer
108 views

Vector Calculus Surface Integral (Limits of Integration)

I'm currently having trouble with the following problem. I believe that I have most of the problem set up, but I am having trouble finding what the limits of integration should be. $\int\limits_S ...
0
votes
2answers
59 views

Line integral over a curve in the II quadrant

I am lost here: $C = x^2 + y^2 = 4$ from $(0,2)$ to $(-2, 0)$. Calculate $ \ \int_c y^2 ds \ \ $ and give reasons the sign is correct. It's obviously the circular arc going counterclockwise from ...
0
votes
1answer
28 views

Question about vector equations of lines and planes

Find the equation of the line going through the point $(2,-3,4)$ ,and which is perpendicular to the plane $ x+2y + 2z = 13$ So I tried this: the normal of the plane is $(1,2,2)$, random point on the ...
0
votes
2answers
51 views

Vectors Question

I have a question regarding Vectors; Find the equation of the plane perpendicular to the vector $\vec{n}\space=(2,3,6)$ and which goes through the point $ A(1,5,3)$. (A cartesian and parametric ...
0
votes
1answer
29 views

How to think about integrals along C.

One of the ways I like understanding things is being able to "see what's going on" so I can hypothesise intuitive results (and then rigorously prove them later). For example, when I see $\int_C fds$, ...
1
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0answers
18 views

The difference between $(\frac{\partial F}{\partial T})_X$ and $\frac{\partial F}{\partial T}$?

Just as the headline says, what is the difference between $(\frac{\partial F}{\partial T})_X$ and $\frac{\partial F}{\partial T}$ ? The former is used at least in thermodynamics, and I find the ...