Tagged Questions

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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1
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1answer
16 views

Showing that a multivariable limit exists

Consider the function $f\colon \mathbb{R^2} \rightarrow \mathbb{R}$ defined on all of $\mathbb{R^2}$ by $f(x,y) = \left\{ \begin{array}{lr} 1, & \text{if } (x,y) \in A\\ 0, ...
0
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0answers
10 views

Aside from work done for a force, is there anything else that $\int_C F\cdot\,dr$ can calculate?

Aside from the work done by a force $F$, is there anything else that $\int_C F\cdot\,dr$ can calculate? I heard that it also calculates "circulation." What other things can be mathematically modeled ...
0
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0answers
12 views

Unitary vector $N$ in the Flux integral

Why in the flux integral $\iint_S F\cdot N\, dS$, the sign of the unitary vector $N$ doesn't matter?
0
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0answers
10 views

How to design own the parametric vector?

I try to design the parametric vector that looks like a roller coaster I know that my equation will like $r(t) = A\sin(t)i + Btj+ C\cos(t)k$, but i want ...
0
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0answers
17 views

Jacobian equals the product of scale factors

I have to prove that in 2 dimensions $J(\frac{x,y}{q_1,q_2})=h_1 h_2$ (1), where $q_1, q_2$ are the new mutually perpendicular coordinates and $h_1, h_2$ are the respective scale factors (exercise ...
3
votes
2answers
60 views

Proving the relation: $∇(\mathbf{u}·\mathbf{v})=(\mathbf{v}·∇)\mathbf{u}+(\mathbf{u}·∇)\mathbf{v}+\mathbf{v}×(∇×\mathbf{u})+\mathbf{u}×(∇×\mathbf{v})$

I have to prove the following relation. I am looking for a solution beyond the obvious brute force method of considering ...
0
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1answer
47 views

Simple Vector Space Rotation

Given Data in the problem & notation convenstions We have 3 rotation vectors called $\vec{\theta_1},\vec{\theta_2},\vec{\theta_3}$, magnitude of these vectors will give you angle of rotation We ...
1
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2answers
50 views

How to denote the inside of a manifold?

In $\mathbb{R}^3$, I want to denote the inside of a closed surface $S$. Now I could define a volume $V$ such that $A = \partial V$, but I do not want to introduce an unnecessary, additional symbol $V$ ...
0
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0answers
23 views

how to find the maximum and minimum value of the directional derivative using Lagrange Multiplier Method?

I want to prove that the maximum value of $\frac{df}{ds}$ is $\left|\nabla f\right|$. To maximize $\frac{df}{ds}$ given by $\nabla f=\frac{\partial f}{\partial x}\hat{i}+\frac{\partial f}{\partial ...
0
votes
2answers
21 views

Dot Product in vector analysis

Suppose I have a vector say v1=(1,2,3) and the dot product of another vector(v2) with v1 is zero. What other information do we need too find v2. I thought this seemed like a pretty trivial question ...
1
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1answer
16 views

how to evaluate $\int \left(\hat{V}\times \frac{d^2\hat{V}}{dt^2}\right)dt$?

if $\hat{V}\left(t\right)$ is a vector function of $t$, find the indefinite integral $\int \left(\hat{V}\times \frac{d^2\hat{V}}{dt^2}\right)dt$ To solve thi first i find for the integrand with ...
3
votes
0answers
72 views

Calculate the flux of $\underline{v}$ across the boundary of the sector.

For $a\in(0,1)$, calculate without use of the divergence theorem the flux of $\underline{v}(x,y) = g(y/x)(-1/x,1/y)$ across the boundary of the sector $ S_a := \{(x,y)\in \Omega : 1\leqslant x^2+y^2 ...
0
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0answers
20 views

Vector Identity

$\displaystyle f=\nabla (\vec{a}.\nabla r^{-1})$ Show that $\nabla.\vec{f}=0$ Given $\vec{a}$ is constant and $r^2=x^2+y^2+z^2$ Attempt: I tried using the identity ...
1
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1answer
15 views

proving an vector indentity in triangle [closed]

P is a point in triangle ABC,to prove that $$S_{\triangle PBC}\cdot \overrightarrow{PA}+S_{\triangle PAC}\cdot \overrightarrow{PB}+S_{\triangle PAB}\cdot \overrightarrow{PC}=\overrightarrow{0}$$ ...
0
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0answers
22 views

Divergence of a radial $1/r$ vector field

Please explain how to obtain the divergence of the function $F(r,\varphi,\theta)=\hat{r}/r$. Is there a solution without computing the surface integral for definition of divergence? Thanks for your ...
0
votes
1answer
31 views

Finding the acceleration

So I am given a problem stated as: a point moves in the plane at speed 1 along the curve $y = x^2$. Find the acceleration at the point (x,y). I know that the velocity is y' = 2x, and that at a ...
1
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2answers
20 views

Hermitian Matrix representation

I know that a Hermitian matrix is a square matrix with complex entries that is equal to its own conjugate transpose, i.e $H = H^{\dagger}.$ But why is $\frac{\partial^2}{\partial x^2}$ Hermitian?
2
votes
1answer
99 views

Jacobian Matrix Requirement for Linear Approximation

It is my understanding that when searching for a linear approximation of a nonlinear function, using the Jacobian (matrix) could help. I did some reading, and read that there is a condition: the ...
0
votes
1answer
13 views

Finding the vector with a given component

When questions goes like "the horizontal velocity is ...", that is referring to the x-component of the vector, correct me if I'm wrong. But I need the vector, is it possible to find the vector from ...
0
votes
1answer
30 views

Asking for help with an inequality

Suppose $u$ is harmonic on an open set $D \subseteq R^n$, then for any $B_R(x_0) \subseteq D$ and $\alpha = ({\alpha _1},...,{\alpha _n}) \in {Z^n}$, $k = |\alpha |$, why the following inequality ...
0
votes
1answer
31 views

Why this equation holds? (the left side is Poisson kernel of a ball)

Let $x=(x_1,x_2,...,x_n)$ be a point/vector on a ball with radius $R$, and $y=(y_1,y_2,...,y_n)$ be a point/vector inside the ball. Let $|x|$ denote the point $x$'s distance to the origin (or the ...
0
votes
1answer
32 views

Vector Cross product - Rearranging issue

Given Data in question I have following relations in vector space$\begin{eqnarray}n_0^{'}(s)=-\kappa(s) \times n_0(s)\\n_1^{'}(s)=-\kappa(s) \times n_1(s)\\n_2^{'}(s)=-\kappa(s) \times ...
1
vote
1answer
33 views

Force field and work

How can I solve the following? Let $F_1=(-y,x,z)$ and $F_2=(y,x,z)$. Calculate for each force field the work done in moving a particle around the circle in the $(x,y)$ plane. Which of the two ...
1
vote
1answer
28 views

Help explaining divergence theorem example

I am looking at an application of the divergence theorem, and I don't understand what's going on. Could anyone explain how to go from the first expression to the second expression (which can then be ...
1
vote
3answers
34 views

A silly question about potential functions

Why have physicists had the idea to define a potential function of a gradient vector field $\vec F$ to be a function $g$ such that $\vec F=-\nabla g$? What changes if we don't put the negative sign? I ...
5
votes
1answer
61 views

Is the helix the unique path with constant curvature and constant torsion?

If a path has both curvature and torsion constant why it is necessarily a helix? How to prove it?
0
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0answers
7 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
69 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
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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
46 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
22 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
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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
47 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
79 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
42 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
32 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
28 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
32 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
22 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
35 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
18 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
54 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
35 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 ...
2
votes
1answer
40 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
73 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
28 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
50 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 ...