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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-2
votes
0answers
17 views

Regular surface/normal line

Let $\mathcal A$ be a regular surface in $\mathbb R^3$ and $P$ a point in $\mathbb R^3\setminus\mathcal A$. Suppose that $C$ is a point at minimum distance from $P$. Show that $P$ belongs to the ...
0
votes
0answers
5 views

Divergence in optional orthogonal coordinate system

I want obtain formula for divergence in optional orthogonal coordinate system(u1,u2,u3) with metric coefficients(h1,h2,h3) with this equation: explain step by step please and attention a lot for the ...
0
votes
2answers
66 views

Proving a cross product satisfies the vector equation

Let vectors $u,v,w \in R^3$ Prove that $u \times (v \times w)$ must be a vector that satisfies the vector equation $x=sv+tw$ where $s,t \in R$ I have no idea where to go with this one, any tips?
0
votes
0answers
31 views

A matrix equation with real coefficients

The problem is the following: Find $\lambda$ such that $ b^{T}A\left[A^{T}A-\lambda L^{T}L\right]^{-1}L^{T}L\left[A^{T}A-\lambda L^{T}L\right]^{-1}A^{T}b-\delta^{2}<0 $ where ...
0
votes
1answer
20 views

Question about velocity vectors

Let $ \vec x(t)$ be a path of class $C^1$ that does not pass through the origin in $R^3$. If $\vec x(t_0)$ is the point on the image of $\vec x$ closest to the origin and $\vec x'(t_0) \ne 0$, show ...
0
votes
1answer
45 views

Exam Question from multivariable calculus.

This question from a previous multivariable calculus exam.I don't know how to start with this question: Let $f$ be differentiable at every point of line segment joining $x_0$ and $x_0+h$.Show that ...
3
votes
1answer
20 views

Vector Calculus intuition: Why is the magnitude of a velocity vector the speed?

From my understanding of basic Calculus (which could very well be completely flawed), the derivative of position with respect to time would give us the slope at every point of that function, which ...
0
votes
0answers
16 views

How to visualize a vector from its components (in spherical coordinates)

Let $$\mathbf{v} = A (1 + \cos \theta) \cos \phi \mathbf{\hat{u}}_{\theta} + A (1 + \cos \theta) \sin \phi \mathbf{\hat{u}}_{\phi}$$ be a vector; $\mathbf{\hat{u}}_{\theta}$ and ...
2
votes
1answer
40 views

Parallel transport along a cardioid

I want to calculate an explicit example of a vector parallel transported along a cardioid to see what happens. Maybe someone could help me with that since no author of any book or pdf on the topic is ...
1
vote
1answer
30 views

Prove $ \nabla \cdot (f \nabla \psi ) = \nabla f \cdot \nabla \psi + f \nabla ^2 \psi $ in general curvilinear coordinates.

Prove $ \nabla \cdot (f \nabla \psi ) = \nabla f \cdot \nabla \psi + f \nabla ^2 \psi $ in general curvilinear coordinates. I have been attempting to do this using general curvilinear dot products ...
0
votes
2answers
73 views

prove $\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$ using index notation

I'm having some trouble using index notation to prove the identity $$\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$$ The closest I can get is by expanding ...
0
votes
1answer
33 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 ...
0
votes
1answer
26 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
38 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
votes
1answer
27 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
vote
1answer
27 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 ...
1
vote
1answer
21 views

Vector Analysis (Parametized curve)

The question is find a familiar parameterized curve that has the property $r(t) \times\dfrac{dr}{dt}=0$. The only curve that I can see that works is the line through the origin. I was just wondering ...
2
votes
1answer
31 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 ...
0
votes
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 ...
1
vote
1answer
51 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. ...
2
votes
4answers
30 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 ...
-1
votes
0answers
28 views

Quick Question: dot product with del

Is $(v \cdot \nabla)F = (\nabla F) \cdot v$? I'm not quite sure.
2
votes
1answer
31 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 $B$ ...
1
vote
1answer
70 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 ...
0
votes
0answers
24 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
46 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
vote
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
votes
1answer
67 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 ...
0
votes
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 ...
0
votes
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
votes
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
vote
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} ...
0
votes
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
votes
1answer
62 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
votes
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 ...
3
votes
2answers
232 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
votes
1answer
34 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
22 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
42 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
votes
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
votes
1answer
25 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
52 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 ...
0
votes
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
64 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 " . ...
0
votes
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
111 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})\\ ...