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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4
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1answer
55 views

Vector Calculus Notation for “Gradient of a Vector”

Given (differentiable) functions $\,n_{1,2}:\mathbb{R}\to\mathbb{R}\,$ we write vector $\renewcommand{\arraystretch}{2}$ \begin{align} \vec{\boldsymbol{n}} = \begin{bmatrix} n_{1} \\ n_{2} ...
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8 views

using Green's Theorem to calculate the Work done for a vector function.

$$f_a(x,y) = \frac {1}{(x^2+y^2)^a}(-y,x)$$ is a vector. Q is a square $$[-1,1]\times [-1,1]$$ and R also a square $$[1,2]\times [-1,1]$$ How do i calculate the Work Integral about Q and R? of the ...
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0answers
24 views

Curl and Potential of Vectorfields.

(-y,x) is a vector$$f_a(x,y) = \frac {1}{(x^2+y^2)^a}(-y,x)$$ After using the curl or rotation formula i get: for $a = 0$ and $a = 1$ there is constant curl or rotation. For the latter the Curl is 0 ...
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0answers
54 views

Proving that a set of points $(x,y,z)$ lie on a line

Let $C$ be the two-sided cone $z^2=x^2+y^2$ 1) Let $S$ be the set of all points ($x,y,z$) in $C$ such that the normal line of $C$ at ($x,y,z$) is perpendicular to the vector $v= \hat i+\hat ...
1
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1answer
61 views

Proof of Kepler's Third Law

Kepler's Third Law states that the square of the time period ($T$) of revolution of a planet about the sun is directly proportional to the cube of the semi-major axis ($a$) of its elliptical orbit. ...
0
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1answer
39 views

Prove H (all m x n real matrices) is a Hilbert space

H consists of all m x n real matrices with addition and scalar multiplication defined as the usual corresponding operations with matrices, and with the inner product of two matrices A, B defined as ...
2
votes
1answer
43 views

Counterexample of linearity of the derivative

I found that the directional derviative $D_xf(0,0) = \sqrt{r^2+s^2} \cdot g \left ( \dfrac{r}{\sqrt{r^2+s^2}}, \dfrac{s}{\sqrt{r^2+s^2}} \right )$ for $x = (r,s)$ and I am then asked to show I ...
1
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1answer
14 views

Finding region for Change of Variables and Double integral problem

I'm running into some trouble on a problem in Vector Calc by Marsden and Tromba. I don't think I am correctly finding the region for my change of variables and the book doesn't have a similar example. ...
0
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1answer
118 views

Proof of Gauss' Theorem in electrostatics using Stokes' and divergence theorems

This was a problem I encountered while solving my homework. PROBLEM:The potential $\phi(x,y,z)$ at any point $P$ due to the charges $q_i, i=1,2,..,n$ with respective position vectors $\vec r_i, ...
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0answers
16 views

How would you express $(\underline{a} \times \underline{b}) \times (\underline{a} \times \underline{c})$ in index notation?

At a guess, I would have said that the answer is $\varepsilon_{ijk}\varepsilon_{jlm}a_{l}b_{m}\varepsilon_{kpq}a_{p}c_{q}$, but I'm guessing that this is incorrect. What is the corrdct expression? ...
1
vote
1answer
23 views

How would you use index notation to show that these vector equations are equal?

How would you use index notation to show that $(\underline{a} \times \underline{b}) \cdot (\underline{a} \times \underline{b}) = |\underline{a}|^{2} |\underline{b}|^{2}-(\underline{a} \cdot ...
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0answers
31 views

Dot product of gradient and tangent vector

Using polar coordinates with variables $r$ and $\theta$. Let $\vec{r}$ be the position vector. Consider $\nabla \theta \cdot \frac{d\vec{r}}{d\theta}$. This is the dot product of the gradient normal ...
0
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1answer
32 views

Showing a rotational symmetry of the Navier Stokes / Euler equation

I'm going through Majda's "Vorticity and Incompressible Flow" and am having a hard time verifying what looks like an easy equality; I suspect its due to my very poor understanding/familiarity of ...
3
votes
3answers
40 views

Shortest distance between point and surface

I wonder if there exists a good enough formula to compute the shortest distance between a point $P=(x_0,y_0,z_0)$ and a surface $\pi$ defined by $F(x,y,z)=0$. There is a lot of simmilar questions in ...
0
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1answer
14 views

Let $f(r)$ be a scalar field from $\Bbb R \mapsto \Bbb R$. Does the Vector field $\vec{F}(\vec{r})=\vec{r}f(r)$ have a potential?

Let $f(r)$ be a scalar function from $\Bbb R \mapsto \Bbb R$. Does the Vector field $\vec{F}(\vec{r})=\vec{r}f(r)$ have a potential? If yes, determine the potential function. ...
0
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1answer
23 views

Calculate the line integral (cyl. coords)

So I have this vector field $$ \textbf{B}=K \left( \frac{\cos \varphi}{\rho^2}\textbf{e}_{\rho}+ \left( \frac{\sin \varphi}{\rho^2}+ \frac{1}{a\rho}\textbf{e}_{\varphi} \right) \right) $$ and the ...
3
votes
1answer
92 views

Counter clockwise flux integral

There's a function $\ \varphi :(0,+\infty) \to \Bbb R$ and another function $\ u:\Bbb R^{2} \to \Bbb R$ defined as $\ u(x,y):=\varphi(x^2+4y^2)$. For some $\ t>0$, let $\ E_{t} = \{{(x,y): ...
0
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1answer
14 views

$\vec{F}= ay\hat{i}+z\hat{j}+x\hat{k}$, if $\int{\vec{F} \vec {dr}}=\pi$, find the value of $a$ for the given curve.

Let $\vec{F}=ay\hat{i}+z\hat{j}+x\hat{k}$ and $C$ be the positively oriented closed curve given by $x^2+y^2=1,z=0$. If $\int{\vec{F}\vec{dr}}=\pi$, then the value of $a$ is : A) $-1$ B) $0$ C) ...
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0answers
59 views

Vector Calculus curl, divergence and gradient identity

Here is the identity I need to prove: $\vec\nabla\cdot(\vec\nabla f\times (\vec r f)) = 0$, where $f$ is a smooth scalar field and $\vec r$ is the position vector. Some tips would be brilliant!
-1
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1answer
26 views

If the curl of $a+b$ is zero. Can one find $b$, knowing $a$?

If I have an equation saying that $\operatorname{curl}(a+b)=0$ and I know $a$ then is there a way to find out $b$? Since $\operatorname{curl}(a+b)=\operatorname{curl}(a)+\operatorname{curl}(b)$, it ...
2
votes
1answer
25 views

Problem with index notaion and einstein summation

I'm in $\mathbb{R^3}$ so $i=1,2,3$ The star is dotproduct. My goal is to simplify this: $$(\hat{e}_i \cdot \nabla)\vec{r} = \frac{\partial \vec{r}}{\partial x_i}$$ And this above I believe is ...
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1answer
18 views

An Equation with Inner Products of Unit Vectors $\{\mathbf v_i : i=0, \cdots, n\}$ Equivalent to $\sum \mathbf v_i = 0.$

I wanna represent the following equation with just inner products of the unit vectors $\{ \mathbf v_i : i=0, \cdots, n\}$: $$\sum _{i=0}^n \mathbf v_i = \mathbf 0.$$ I had thought following ...
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2answers
35 views

Parametric curve: $x=\frac{a}{2}(t+\frac{1}{t})$, $y=\frac{b}{2}(t-\frac{1}{t})$?

What kind of shape is the parametric curve described by: $$x=\frac{a}{2}(t+\frac{1}{t})$$ $$y=\frac{b}{2}(t-\frac{1}{t})$$ $a,b \in\mathbb{R^+}$ ?
1
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1answer
45 views

How to calculate the line integral and substitute $dx\,\ dy$ in a question on Green's theorem

The question states that: Verify Green's Theorem on the plane for $\oint_C (2x-y^3)dx-xydy$ where C is the boundary of the region enclosed by the circles $x^2+y^2=1$ and $x^2+y^2=9$. My attempt: ...
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0answers
22 views

Parabolic coordinates as functions of $x$, $y$ and $z$

OK, so I have this problem: Express the parabolic coordinates $\textit{u}$, $\textit{v}$ and $\varphi$ as functions of the cartesian coordinates $x$, $y$ and $z$, how does $\textit{u}$, $\textit{v}$ ...
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0answers
30 views

What does symmetry imply about the solution in mathematics? (Example: Gauss' law)

Suppose you have an infinite cylinder and are considering a field $\mathbf{D}$ caused by physical elements within the cylinder such that it satisfies $\int \mathbf{D}\cdot d\mathbf{a} = Q_{free}$. ...
0
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1answer
27 views

Calculate the unit vectors in the curvilinear coordinate system

Calculate the unit vectors in the curvilinear coordinate system and show that they are orthogonal $$u_1=x^2-y^2 \\ u_2=xy \\ u_3=z$$ I'm not really sure how to do this. I know that I need the ...
1
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1answer
11 views

Snider Section 4.7 Problem 10.

Calculate $\int \int \vec{F} \cdot d\vec{S}$ over the section of surface surface is $x = u^2$, $y = uv$, $z = \frac{1}{2} v^2$ bounded by the curves $u = 0$, $u = 1$, $v = 0$, and $v = 3$, for the ...
0
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1answer
25 views

$a^Tx=b$ denotes a hyperplane. Is it always true that $a//x$

$a^Tx=b$ denotes a hyperplane. Is it always true that $a//x$? $x \in R^n$ $a\in R^n$ anb $b \in R$.From this data can we infer that $\vec a$ is parallel to $\vec x$ and if yes why?
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2answers
36 views

How to find the potential function of a vector field?

I calculated that $\frac{dP}{dy} = \cos(y) = \frac{dQ}{dx}$ This tells me that the potential function exists, however I can't figure out what it is. So far I have found that $\frac{df}{dx} = ...
2
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1answer
64 views

Verification of Green's Theorem homework help

QUESTION: Verify Green's Theorem for $$\oint (x^2-2xy)dx+(x^2y+3)dy$$ around the curve $y^2=8x$ and $x=2$. My attempt: L.H.S. comes out to be $\frac{128}{5}$ which is correct acc. to the book. For ...
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vote
3answers
35 views

Given (X, ||•||) normed space, prove that only X itself and empty space are clopen.

I' d like to ask you for some help. I' ve to prove the problem stated in title, but without using the knowledge that normed space is connected.And I just got no idea how to do so... Thanks for any ...
0
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1answer
16 views

Relating existence of a “potential” with exactness of a certain form

Let $\Omega \subseteq \Bbb R^2$ be an open set, and let $\omega = \omega_1\,{\rm d}x_1 +\omega_2 \,{\rm d}x_2$ be a $1$-form in $\Omega$. Consider the field: $$L = \omega_2 \frac{\partial}{\partial ...
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2answers
22 views

Differentiating $\vec{r}(t)=(3\sin\frac{t}{t_0},4\frac{t}{t_0},3 \cos \frac{t}{t_0}); \space \space t\in \Bbb R$

I want to differentiate the following vector with respect to $t$ $$\vec{r}(t)=(3\sin\frac{t}{t_0},4\frac{t}{t_0},3 \cos \frac{t}{t_0}); \space \space t\in \Bbb R$$ Can I treat $t_0$ like a constant ...
1
vote
1answer
28 views

An exercise/solution-based reference on vector calculus

I need to remember several things from vector calculus (line integrals, surface integrals, vector fields, Green's, Stock's & other famous theorems, etc.). Towards this, I'm looking for a resource ...
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1answer
21 views

Let $S$ be the sphere of radius $2$ and $C$ be the circle obtained by intersecting $S$ with the plane $\{x-y+z=3\}$. Parametrize $C$.

Let $S$ be the sphere of radius $2$ centered at the origin in $\mathbb{R^3}$. Let $C$ be the circle obtained by intersecting $S$ with the plane $\{x-y+z=3\}$. Parametrize $C$. If I let $y=x+z-3$ and ...
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0answers
106 views

Stokes' Theorem and Vector Fields with Jump Discontinuities

What are the continuity requirements on a vector field $\boldsymbol{A}$ such that Stokes' theorem, $$ \iint_S\nabla\times\left[(\boldsymbol{\hat{x}}\cdot\nabla\phi)\boldsymbol{A}\right]\cdot ...
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votes
1answer
21 views

Area of parametric surface (theory)

In the picture below $\left \|\Delta u_i r_u \times \Delta v_i r_v \right \|$ is the area of the parallelogram $\Delta T_i$ Can someone please explain why the sides of the parallelogram $\Delta T_i$ ...
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0answers
58 views

Surface area common to two perpendicularly intersecting cylinders

I need help to calculate the following surface area: the surface area common to the two cylinders $x^2 + y^2 = a^2$ and $x^2 + z^2 = a^2$ using surface integrals essentially. My attempt: Let ...
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0answers
14 views

Evaluating gradiant of vector basis r in Spherical Co-ordinates

If I were to define $\nabla$ in spherical polar co-ordinates: $$ ...
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0answers
23 views

Net flux zero equivalent to vanishing solution?

Consider \begin{align*} \Delta u(P) = 0, \quad P\in \Omega,\\ \frac{\partial u(P)}{\partial n_{p}} = f(P),\quad P\in \Gamma, \end{align*} where $\Omega$ is the infinite domain in the plane outside ...
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1answer
30 views

Helmholtz Decomposition in Sobolev Space

Let $\Omega$ is a $C^2$-smooth bounded domain in $R^2$ (not assume to be simple connected) then $f\in W^{1,2}(\Omega) $ can be represented as: $$f=\nabla^{\perp}b+\nabla\varphi$$ With $b,\varphi\in ...
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0answers
19 views

Divergence from rate of fluid volume expansion.

Problem: The divergence of $\vec{F}$ is the time rate of change of the volume of a rectangular parallelepiped of body fluid, per unit volume, as the size of the box goes to zero. Show this. [Hint: ...
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0answers
15 views

Estimating certainty of angle for rotations using curl filters.

Background Inspired by this answer which manages to make a connection between (the angle of) a proper rotation (displacement field of a rotation) and the curl of a vector field. This question aims to ...
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1answer
59 views

Surface Integral over a cone

I need help to find the solution to the following problem: $$ I=\iint_S \vec A \cdot\ d \vec s$$ over the entire surface of the region above the $xy$-plane bounded by the cone $x^2 + y^2 = z^2$ and ...
0
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1answer
25 views

How $\nabla_x f(x,y) = -2y$ holds when $f(x,y) = 2x \cdot y$

Let $x = (x_1, x_2, x_3), y = (y_1, y_2, y_3)\in\mathbb R^3$ and $f(x,y) = 2x \cdot y$. It is written in the paper that $$ \nabla_x f(x,y) = -2y. $$ I cannot understand how it holds. I think it is ...
1
vote
1answer
24 views

dot-product spherical

I want to calculate dotproduct of $ e_r *e_\phi $ they are unit vectors in spherical. where my spherical coordinates is $(r,\phi,\theta)$ My attempt is first to convert them to cartesian: which ...
1
vote
2answers
71 views

Gauss Divergence theorem gives a wrong result for a surface integral

Evaluate $\iint _{{S}}(y^{2}z^{2}i+z^{2}x^{2}j+x^{2}y^{2}k)\cdot \,ds$ where $S$ is the part of the sphere $x^{2}+y^{2}+z^{2}=1$\, above the xy-plane. Answer to this question is $\pi/24$ as ...
1
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1answer
93 views

Prove $\nabla u \times \nabla v =0$ is a necessary and sufficient condition that $u$ and $v$ are functionally related by the equation $F(u,v) = 0$

Let $u$ and $v$ be differentiable functions of $x$, $y$ and $z$. Show that a necessary and sufficient condition that $u$ and $v$ are functionally related by the equation $F(u,v) = 0$ is that ...
5
votes
2answers
139 views

How does curl relate to rotation?

The operation mathematically means $$(\nabla \times \vec A)\cdot\hat n = \lim_{\Delta S\to\ 0} \frac{\oint\vec A\cdot\ d\vec l }{\left | \Delta S \right |}$$ and the proof of this is quite logical. ...