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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1answer
30 views

Integration by parts on all of $\mathbb{R}^n$ with $n>1$

So this came up as I was thinking about the uniqueness of solutions to the wave equation. I have seen proofs for uniqueness on all of $\mathbb{R}$ or on bounded subsets of $\mathbb{R}^n$, but never ...
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1answer
22 views

Rotation matrix in R^3

Rotation matrices in $R^3$ are given by $$S = \begin{pmatrix} \hat e'_1.\hat e_1 & \hat e'_1.\hat e_2 & \hat e'_1.\hat e_3 \\ \hat e'_2.\hat e_1 & \hat e'_2.\hat e_2 & \hat e'_2.\hat ...
1
vote
1answer
36 views

Differentiation of $xx^T$ where $x$ is a vector

How is differentiation of $xx^T$ with respect to $x$ as $2x^T$, where $x$ is a vector? $x^T $means transpose of $x$ vector.
3
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1answer
25 views

How to find these quantities so as to conform to these conditions?

Suppose $a \in \mathbb{R}^k$, $b \in \mathbb{R}^k$. Then how to find $c \in \mathbb{R}^k$ and $r > 0$ such that the following holds? For any $x \in \mathbb{R}^k$, we have $$|x-a| = 2 |x-b|$$ if ...
1
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1answer
26 views

Intuition behind divergence?

$\overrightarrow f = 3x\overrightarrow i - 3y\overrightarrow j$ $\overrightarrow g = 3x\overrightarrow i + 3y\overrightarrow j$ If I calculate the divergence of $f$ I get $0$. If I calculate the ...
2
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1answer
28 views

Vectors and polyhedra: a surprising fact

Given a $n$-faced polyhedron, associate to each face an outward-pointing normal vector with length equal to the area of that face. Show that the sum of these $n$ vectors is zero. I've already proved ...
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1answer
17 views

I want to know where I did wrong in finding the plane equation

I am asked to give 3 plane equation where the third plane will passes through the intersection of the first 2 planes and parallel to y axis. I came up with 2 plane equation which is also parallel to ...
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1answer
49 views

Why does the arc-length formula have form $\int_a^b\left|\left|\frac{d\vec{f}(t)}{dt}\right|\right|_2dt$ for C1 curves?

This discussion focuses on $\mathcal{C}^1$ curve on $\mathbb{R}^n$. But feel free to talk about the case where we only have a continuous curve or the scenario with a manifold with a metric in general. ...
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4answers
25 views

Dot Product of a Non-Zero vector with a Null Vector

The dot product of two vectors let us say $\vec{A}$, and $\vec{B}$ is defined as $$\vec{A} \cdot \vec{B} \equiv AB\cos\theta,$$ where $A$ and $B$ are the magnitudes of the vectors $\vec{A}$ and ...
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0answers
28 views

Quick Question: dot product with del

Is $(v \cdot \nabla)F = (\nabla F) \cdot v$? I'm not quite sure.
2
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1answer
29 views

Solve a problem using vectors

The purpose of this problem is to use vectors to show that the medians of a triangle all meet at a point. First, I have to show that $P$ (see the picture below) must lie two-thirds of the way from $B$ ...
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1answer
69 views

How to expand the term $\nabla \times (\mathbf{A} \times \nabla)$

Using Feynman notation $\nabla \times (\mathbf{A} \times \nabla) =\nabla_A \times (\mathbf{A} \times \nabla) +\nabla_\nabla \times (\mathbf{A} \times \nabla)$ but I have a problem while expanding the ...
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0answers
20 views

Curl and divergence of vector fields

Does the vector operations of fields, like curl and divergence require the field to be defined at the point or only in the nbd of the point as the definition of curl is So is the differential ...
0
votes
1answer
45 views

A proof involving vectors

This problem concerns three circles of equal radius $r$ that intersect in a single point $O$. Let $W_1,W_2,W_3$ denote the centers of the three circles and let $\vec w=\overrightarrow {OW_i}$ for ...
1
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1answer
19 views

Proving cross product identities

In my textbook, the author claims that the following can be proved by chaining vector triple product and scalar triple product $$\text{i.) }(A \times B) \cdot (C \times D) = (A \cdot C)(B \cdot D) - ...
5
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1answer
59 views

Geometric proof for triple vector product Jacobi identity

I believe the vector identity $\vec{a} \times (\vec{b} \times \vec{c}) + \vec{b} \times (\vec{c} \times \vec{a}) + \vec{c} \times (\vec{a} \times \vec{b}) = 0$ is called the Jacobi identity and I know ...
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2answers
18 views

Curl and gradient properties for $f ( r)\vec r$

I need to show that the curl of $f( r) \vec{r}$ is $0$. I think I can use this property: $$\operatorname{curl}(Av) = \operatorname{grad}(A)\times v+A \operatorname{curl}(v)$$ I have started ...
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0answers
23 views

Integral Inequality involving the Euclidian Norm

I have spent several hours trying to establish the inequality shown in the attached photo. Here we assume that $\vec{r}(t)$ is a vector function in $R^n$, and is integrable on $[a,b]$. I am in need of ...
0
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1answer
24 views

Restriction of a div-free vector field to a plane?

Suppose we restrict a divergence-free vector field on $\mathbb{R}^3$ to some plane. What can be said about the restricted vector field? It no longer has to be divergence-free, of course. But can ...
1
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2answers
23 views

On the Definitions of Line and Surface Integrals

When motivating the definitions of line and surface integrals, one usually defines the length and area elements \begin{align*} ds &:= \| \vec{r}^{\, '}(t) \| dt, \\ dA &:= \| \vec{\Sigma}_{u} ...
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1answer
28 views

Vectors - collinear and perpendicular

A bird is at point P whose coordinates are (4, -1, 5)m. The bird observes two points $P_1$ and $P_2$ having coordinates (-1,2,0) and (1,1,4) respectively. At time t = 0, it starts flying in the plane ...
0
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1answer
61 views

Calculate integral involving product of curls

I want to show that $ \frac{1}{k^2} \int \, (\nabla \times \vec M)\cdot (\nabla \times \vec M)^* d^3 r= \int \vec M \cdot \vec M^*d^3 r$. $\vec M$ is defined as $\vec M_{jm} = i\frac{m}{\sin ...
0
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1answer
16 views

Diagonalization of matrix using change of variables

In linear algebra, we know that a system of equations $AX=b$ can be easily solved if $A$ is found to be of diagonal nature. If however $A$ is not diagonal but can be changed into a diagonal form by ...
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2answers
224 views

Proof of trigonometric identity using vector calculus

Question: Using vector calculus, show that $\sin (A+B) = \sin A \cos B + \cos A \sin B$ I have no idea how to even attempt the question. A small hint to help me get started would be greatly ...
2
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1answer
33 views

Integration against divergence free vector fields

Let $\chi:\Omega\to \mathbb{R}^n$ be a vector field on a bounded, smooth domain $\Omega \subset \mathbb{R}^n$. Assume that for any divergence free vector field $\eta:\Omega \to \mathbb{R}^n$ we have ...
0
votes
1answer
21 views

finding volume of the cone by using the dot product

Vector u = i +j -2 k and v = 2 i + 3 j can be used to form a cone. The cone forms by rotating u about v through 360 degrees. Find the volume of the cone. I drew the diagram already, I need the r and ...
0
votes
1answer
40 views

Gradient of real part = real part of gradient?

Suppose f(x,y,z) maps $\mathbb{R}^3\rightarrow\mathbb{C}^1$. That is, it takes in three real numbers and spits out a complex number. Does the following always hold: $$\vec\nabla ...
0
votes
1answer
18 views

Rotating vector functions

Suppose you have a vector function of space: $\vec{E}(x,y,z)=Ex(x,y,z)\hat{x}+Ey(x,y,z)\hat{y}+Ez(x,y,z)\hat{z}$. Suppose now you want to rotate the whole vector function by using a unitary rotation ...
0
votes
2answers
31 views

dot product with unknowns of the vectors

$a = (8 , y)$ $b = (2, 3)$ $c = (x, y)$ if $a \cdot c = 10$ and $b \cdot c = 8$, find the values of x and y. I did up to $10 = 8x i + y^2 j$ $8 = 2x i + 3y j$
10
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1answer
246 views

Intuition behind curl identity

Is there any clear intuition behind the identity $$ \nabla\times (\nabla\times A)=\nabla (\nabla \cdot A)-\nabla ^2A $$ Though the result is useful and not difficult to derive, it doesn't quite ...
0
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1answer
24 views

Calculating the divergence of a central vector field.

I am trying to calculate the divergence of a central electric field, namely the electric field due to a point charge and my book begins like this: http://imgur.com/bW9tPEZ However in the last line I ...
0
votes
1answer
51 views

What am I doing wrong in this volume integral (divergence theorem)?

I'm learning about the divergence theorem. If I have a vector function $f(x,y,z)=\sqrt {x^2+y^2} \cdot (x,y,z)$ and I want to get $\iint\limits_A f(x,y,z) \, d A $ (easy to evaluate, but I thought I'd ...
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0answers
24 views

Laplacian in arbitrary spherical coordinates?

assume that $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is a function that just depends on the radius vector $r$, so no angular dependence(!). Can we say what $\Delta f$ is in arbitrary coordinates $r ...
0
votes
2answers
61 views

Definite integral-dot product

I have an integral equation containing dot product $$\int_{0}^{L} \left(\frac{a}{L}.b(s)\right)\mathrm ds\tag 1$$ Data Given a is a constant vector of size 3 b(s) is a varying vector of size 3 " . ...
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0answers
36 views

Farthest vector direction relative to other vectors

I wished to know the cheapest computational means (be it analytical or numerical) to find the vector from origin (normalised or not; I do not care about its magnitude) given any arbitrary set of ...
2
votes
2answers
48 views

What is the simplest way to understand $\nabla \cdot (\nabla \times \textbf{A} ) =0$?

Of course one can prove it by brute force. But how to Understand it, in the simplest way? Another issue is $\nabla \times (\nabla f) =0 $.
4
votes
1answer
57 views

Spherical integral

Let $y \in \mathbb{R}^n$ be fixed. Is there a nice expression for the following integral taken over the unit sphere in $\mathbb{R}^n$? $$ \int_{\|x\|=1} e^{2\pi i (x \cdot y)}~dx $$
3
votes
2answers
109 views

$\Delta \vec{v}=0$ implies $\nabla\cdot \vec{v}=\nabla\times \vec{v}=0$?

\begin{align} \Delta\overrightarrow{v}&=\nabla(\nabla\cdot\overrightarrow{v})-\nabla\times(\nabla\times\overrightarrow{v})\\ ...
0
votes
0answers
50 views

Plotting parametric form of a gradient

This is driving me batty. I'm trying to figure out how to plot the gradient of a circle function (is that a vector field?) in parametric form. I don't understand what values to plug in to a get a ...
0
votes
1answer
26 views

Why is the derivative Df(p) defined to $ \in \Lambda^1 (\mathbb{R}^n) $, or how is it a 1-form?

I know that obviously the differential operator D would be a differential form through the word differential. But in Spivak Calculus on Manifolds he defines a k-form w $ \in \Lambda^k ...
1
vote
2answers
92 views

Using Stokes theorem to integrate $\vec{F}=5y \vec{\imath} −5x \vec{\jmath} +4(y−x) \vec{k}$ over a circle

Find $\oint_C \vec{F} \cdot d \vec{r}$ where $C$ is a circle of radius $2$ in the plane $x+y+z=3$, centered at $(2,4,−3)$ and oriented clockwise when viewed from the origin, if $\vec{F}=5y ...
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vote
2answers
87 views

derivative formula $\nabla \times (\mathbf{a} \times \mathbf{r}) = \nabla \cdot(\mathbf{a} \wedge\mathbf{r}) = (n-1)\mathbf{a}$

Assume $\mathbf{r}=\mathbf{x}−\mathbf{x}′$ is the position vector in $\mathbb{R}^n$, for constant $\mathbf{a}$, we have $$\nabla \times (\mathbf{a} \times \mathbf{r}) = \nabla \cdot(\mathbf{a} ...
2
votes
2answers
90 views

derivative $\nabla \frac{\mathbf{r}}{r^k}$ in the context of Geometric Calculus

Suppose $\mathbf{r} = \mathbf{x - x'}$ is the position vector in $\mathbf{R^n}$, and $r = |\mathbf{r}| = |\mathbf{x - x'}|$. Do we have $\nabla \frac{\mathbf{r}}{r^k} = \frac{n-k-1}{r^k}$ or $\nabla ...
0
votes
0answers
14 views

Method of Characteristics (Change of Co-ordinates)

Here below is the notes about the change of co-ordinates from $xy$-plane to $\xi\eta$-plane. I wanna ask for why dot product works for the change, i.e. $\xi=(x,y) \cdot (a,b)$ and $\eta=(x,y) \cdot ...
1
vote
1answer
22 views

How to calculate length and area for this curve?

$C : x^{2/3} + y^{2/3} = 1$ I'm stuck, so any tip will be helpful Thanks in advance!
0
votes
0answers
25 views

How to calculate the flow of fluid through this closed surface?

Considering a fluid whose velocity field is $\vec{v}(x,y,z)= (y^{3}e^{-z^{2}} + x)\vec{i} + (ze^{x} + y^{2})\vec{j} + (cos(x^{2}+ y^{2}) +2z)\vec{k}$ Calculate the flow of fluid through the closed ...
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2answers
35 views

geometry proof with triangles using vector

in a triangle ABC, P, Q are points on AB and R, S are points on BC such that AP=PQ=QB and CR=RS=SB. Show that PR bisects AS.
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1answer
46 views

How to calculate this area? (portion of a sphere inside a cylinder )

The area of ​​the portion of the sphere $ x^{2} + y^{2} +z^{2} = 1$ located inside of the cylinder $x = x^{2} + y^{2}$, and above the plane $z = 0$. I'm stuck, so any tip will be helpful Thanks in ...
2
votes
1answer
16 views

On the Continuity of the Jacobian of a diffeomorphism

Let $\phi:U\longrightarrow V$ be a diffeomorphism between the open sets $U, V\subseteq \mathbb R^n$. Provided $J\phi(x)\neq 0$ for all $x\in U$ we have a map $$J\phi:U\longrightarrow GL_n(\mathbb R), ...
0
votes
1answer
47 views

Flow of fluid through a really tricky closed surface S (divergence theorem)

Considering a fluid whose velocity field is $\vec{v}(x,y,z)= (y^{3}e^{-z^{2}} + x)\vec{i} + (ze^{x} + y^{2})\vec{j} + (cos(x^{2}+ y^{2}) +2z)\vec{k}$ Calculate the flow of fluid through the closed ...