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
56 views

Infinity as a boundary condition - Laplace's equation

I am familiar with the uniqueness theorem that claims that given conditions on the boundary of a closed surface, if there exists a function $\varphi$ such that $\nabla^2\varphi = 0$ which also satisfy ...
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0answers
15 views

What would the phrase “attain an upper bound of the line integral” mean for vector fields?

I am working on an exercise that is asking to find the upper bound of a line integral over the unit disk where the vector field has magnitude one. I am then asked to find a vector field that attains ...
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1answer
55 views

Integral Inequality for Bound on Gradient of Solution to Heat Equation

My overall aim is to show that, for a bounded solution $u(x,t)$ to the heat equation in $\mathbb{R}^n \times [0,T]$ with boundary condition $u(x,0) = f(x)$, $$\max |\nabla u(x,T) | \leq \frac ...
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2answers
47 views

linear transformation matrix under the line integral

Is there a general methodology/approach for evaluating an integral of this form? $$ \int_C {\bf Ax} \cdot \mathrm{d}{\bf x} $$ Assume ${\bf x} \in \mathbb{R}^{n \times 1}$ and ${\bf A} \in ...
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0answers
55 views

Find a vector field G such that F = curl(G), Given F = …

Given $F = (xe^y, -x \cos z, -ze^y)$, I find that $\operatorname{div}(F) = 0$ thus there should exist a vector field $G$ such that $\operatorname{curl}(G) = F$. I run into issues finding the solution. ...
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0answers
17 views

Stokes' theorem and line integrals

Use Stokes's theorem to show for $\mathbf{F} = -y\mathbf{i} +x\mathbf{j}+z\mathbf{k}$ where the surface $S$ is the southern hemisphere of the unit sphere, i.e. $S$ is defined by $x^2+y^2+z^2=1$, ...
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1answer
86 views

Is the Laplacian a vector or a scalar?

Need to prove $\operatorname{div}(\nabla u)=\nabla ^2 u$ where $u=g(x,y,z)$ The RHS is the Lapacian which we were told is a vector. But $\nabla u=(g_x,g_y,g_z)$ and the divergence of that is ...
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1answer
28 views

Area of surface [duplicate]

What is the area of the region that is bounded by the curve $$\vec{R}(t)=(\cos^3t, \sin^3t), 0\leq t<2\pi?$$ I have no idea how to start here or what i have to use.
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1answer
37 views

Why is the rotation not zero and the divergence zero in the figures below?

Why is the rotation not zero and the divergence zero in figure 1 and figure 2 below? Figure 1 Figure 2
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0answers
39 views

Immersion except at the origin. [closed]

Whitney showed that for maps of two-manifolds into $\mathbb{R}^3$, a typical cross cap looks like the map $(x, y) \to (x, xy, y^2)$. Prove that this is an immersion except at the origin.
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1answer
26 views

Divergence of $\phi$ from p

I am reading a paper which is based mostly on divergence. I tried to get a basic understanding of divergence but I cannot see how it is linked with this aspect. It says: $D(\phi,p) = \phi . ...
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4answers
47 views

Difference between $\nabla T$ and $\nabla \cdot E$

Why is $\nabla T = (\frac{\delta T}{\delta x},\frac{\delta T}{\delta y},\frac{\delta T}{\delta z})$, but $\nabla \cdot E \neq (\frac{\delta E}{\delta x},\frac{\delta E}{\delta y},\frac{\delta ...
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3answers
48 views

Why is $a\times b = 0$ when $b = 2*a$?

In vector calculus, why is $a\times b = 0$ when you know that $b=2*a$? So how how do you know that a crossproduct of a vector and two times that vector is always zero?
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1answer
27 views

Integral of 2-D Laplacian

I am so confused on these integrals. Here is the question. Problem $$G(x,y)=\ln(x^2+y^2)/2$$ Calculate the 2-D Laplacian $\Delta^2G$ For the interior $D$ of the circle $C$ of radius $a$ calculate ...
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0answers
18 views

Calculating the Flux of a Surface

I am having some trouble with this problem, I feel like I am just confusing myself and I could really use some direction. Problem "For positive $a$ and $h$ let $A$ designate the region of $R^3$ ...
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1answer
17 views

Under what condition does $A^T(B \times C) + (B\times C)^T A = 2A^T(B \times C)$, A,B,C vectors

In my classical mechanics text book there is a formula that states $(\dot r_c + \omega_i \times d_i)^T (\dot r_c + \omega_i \times d_i)$ give rise to $\dot r_c^T \dot r_c + 2\dot r_c^T(\omega_i ...
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0answers
26 views

Find a vector function that represents the curve of intersection of $x^2+y^2=4$ and $z=x$

Find a vector function that represents the curve of intersection of $x^2+y^2=4$ and $z=x$ I changed $x^2+y^2=4$ to $4sin^2\theta + 4cos^2\theta = 4$ so $x=2cos\theta$ and $y=2sin\theta$ and then ...
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1answer
35 views

Closed surface integrals

Can somebody give me hints to solve the following question? I need to find the closed surface integral (using divergence theorem) of $$\oint \vec{r} (\vec{a} \cdot \vec{n}) da$$ where $\vec{n}$ is ...
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2answers
27 views

What will be value of $\vec{r} \cdot \nabla$

I was studying on Nabla Operator and saw that $\nabla \cdot \vec{r} \neq \vec{r} \cdot \nabla$ So, if I were to find $\vec{r} \cdot \nabla$ how would I calculate it? I know that $\vec{r} \cdot ...
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2answers
41 views

Divergence and Curl (involving constant vectors)

How find the divergence and Curl of the following: $(\vec{a} \cdot \vec{r}) \vec{b}$, where $\vec{a}$ and $\vec{b}$ are the constant vectors and $\vec{r}$ is the radius vector. I have tried solving ...
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1answer
29 views

Divergence and Curl of the vectors

How to find the divergence and the curl of the given vectors? a. $( \vec{u} \cdot \vec{r}) \vec{v}$ b. $( \vec{u} \cdot \vec{r}) \vec{r}$ c. $( \vec{u} \times \vec{r})$ d. $ \vec{r} \times(\vec{u} ...
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0answers
27 views

Two methods of calculating a Jacobian determinant

Suppose you have two fluid bodies, one described by a set of vectors $V$, and a perturbation of $V$ given by $V+\Delta V$. Suppose that the two regions are related by the transformation $\mathbf ...
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0answers
20 views

Derivation of centrifugal acceleration with Coriolis theorem

Is there a way to derive the centrifugal acceleration of an object rotating with a constant speed $V$ on a circle with radios $r$ with Coriolis theorem ?? thanks
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0answers
40 views

Index of a zero of a normal vector field

Let $M$ be a manifold and $S\subset M$ a submanifold. Assume whatever regularity you need (smoothness, compactenss, orientability, empty boundary, proper embedding of $S$,... this question is ...
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1answer
18 views

Need Help look at function continuity

Consider the following piece-wise function: $$f(x, y) = xy \frac{x^2-y^2}{x^2+y^2}$$ for $(x,y) \neq (0,0)$ and $f(0,0)= 0$. Discuss the continuity of $f$ at $(0, 0)$. Calculate $\partial f / ...
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1answer
56 views

If $\alpha$ is a unit speed curve of constant curvature lying in a sphere, then $\alpha$ is a circle.

I'm trying to solve the following problem but got stuck along the way. I would like some help on getting this through. Prove that if $\alpha$ is a unit speed curve of constant curvature lying in a ...
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2answers
31 views

If $\vec p\times((\vec x-\vec q)\times\vec p)+\vec q\times((\vec x-\vec r)\times\vec q)+\vec r\times((\vec x-\vec p)\times\vec r)=0$, find $\vec x$

Let $\vec p, \vec q$ and $\vec r$ are three mutually perpendicular vectors of the same magnitude. If a vector $\vec x$ satisfies the equation $\begin{aligned} \vec p \times ((\vec x - \vec q) \times ...
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3answers
101 views

Let $\alpha(s)$ be a unit speed curve in $R^2$. Show $\kappa=|\frac{d\theta}{ds}|$

I'm lost on solving the following problem. Let $\alpha(s)$ be a unit speed curve in $R^2$. Show $\kappa=|\frac{d\theta}{ds}|$, where $\theta$ is the angle between the positive $x$-axis and the ...
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3answers
57 views

Finding a point a certain distance away from 2 points

I need to find a point that is a certain distance away from two known points. Where $P_1, P_2, L_2$ and $L_1$ are all defined and that is all that is known. How do I find $P_3?$ Kind Regards.
2
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1answer
53 views

Proving vector Identities (Using the Permutation Tensor and Kroenecker Delta)

Prove the following vector identities by using permutation tensor and kroenecker delta. $$(\vec{A} \times \vec{B}) \times (\vec{C} \times \vec{D}) = (\vec{A} \cdot (\vec{B} \times \vec{D})) \cdot ...
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2answers
47 views

Question in vector algebra regarding minimum value of modulus.

If $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are three coplanar unit vectors such that $\vec{a} +\vec{b} +\vec{c} =0$. If three vectors $\vec{p}$ , $\vec{q}$ , $\vec{r}$ are parallel to $\vec{a}$ , ...
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0answers
19 views

Using Stokes's theorem

Use Stokes's theorem to show for $\mathbf{F} = -y\mathbf{i} +x\mathbf{j}+z\mathbf{k}$ where the surface $S$ is the southern hemisphere of the unit sphere, i.e. $S$ is defined by $x^2+y^2+z^2=1$, ...
2
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1answer
80 views

What are some good books on vector analysis in higher dimenesion

What is some good books specifically on vector analysis in higher dimension? Standard vector calculus book usually only introduced double and triple integral method
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1answer
37 views

using the divergence theorem

Use Gauss Divergence Theorem to compute $$\int \int \limits_S F\cdot n \,dS$$ where $n$ is the outward normal for the following: $S$ is the surface $x^2 +y^2=z^2$ and $z \in [0,1]$, and ...
1
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1answer
45 views

Convert a general equation system to matrix form

I have an equation, where $x$ is a vector of variables and the rest (vector $a$, vector $b$, matrix $M$ and $c_1, c_2, c_3$) are parameters: $a_i - x_i + c_1\sum_{j=1}^{n} m_{ij} x_j - c_2 ...
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3answers
641 views

Is Stokes' Theorem natural in the sense of category theory?

Stokes' Theorem asserts that for a compactly-supported differential form $\omega$ of degree $n-1$ on a smooth oriented $n$-dimensional manifold $M$ we have the marvellous equation $$\int_M d\omega = ...
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3answers
87 views

Why is Green's theorem asymmetric in $x$ and $y$?

Green's theorem is $$\oint_{\partial D} (P\, dx+Q\, dy) = \iint_D dx\,dy \: \left ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)$$ where one can see that the RHS is ...
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2answers
43 views

Solution of an integral containing vectors

I'm currently trying to solve the integral: $$ I(\vec{a},\vec{b})=4\pi\int\limits_0^1\frac{\mathrm{d}u}{1-(\vec{a}u+\vec{b}(1-u))^2}, $$ but I can't seem to find a good starting point. I know that if ...
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0answers
42 views

Does anybody know how to actually derive spherical harmonics in a way that is historically accurate and intuitive?

And by "historically accurate", I mean without resorting to techniques of derivation which were developed after the fact or explanations which use the very concept they're trying to explain. The few ...
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0answers
46 views

Simple Vector Calculus Integral

A little stuck on what I assume is probably a fairly basic vector calculus integration problem, but I haven't been able to find any resources online that deal with multivariable integration in a way ...
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33 views

Derive the equation of first variation for a flow of a vector field.

This is a problem from Susan Colley's Vector Calculus. I have trouble understanding the solution to it. Problem: Derive the equation of first variation for a flow of a vector field. That is, if ...
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2answers
47 views

Wording on this curl question

Consider the scalar field defined below: $$f:\mathbb{R}^3 \rightarrow \mathbb{R}^3, \hspace{2mm} F(x,y,z)=(x^2y^3,xy,xz^4)$$ Find the curl of $f$ at each point where it exists. I am a bit confused on ...
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2answers
14 views

cuboid with $z=0$ and $z=y$

Compute $\int \int _S F \cdot n \hspace{2mm} dS$ where $$F(x,y,z)=(x-z\cos y, y-x^2+x\sin z+z^3, x+y+z)$$ and $r$ is the surface that bounds the solid between the planes ...
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0answers
22 views

divergence theorem cube question

Compute $$\int \int _S F \cdot n \hspace{2mm} dS$$ where $S$ is the surface of the cube bounded by the six planes $$x=0,\hspace{2mm}x=2,\hspace{2mm}y=0,\hspace{2mm}y=4,\hspace{2mm}z=0, \hspace{2mm} ...
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2answers
49 views

Does the trace and determinant uniquely determine the eigenvalues of a 3 by 3 matrix with algebraic multiplicity of 2?

I have a 3 by 3 matrix $M$ whose eigenvalues are $a$, $b$, and $b$. The determinant and trace of $M$ are known from its eigenvalues: $det(M)=ab^2$ and $Tr(M)=2b+a$. I wanted to show that if ...
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3answers
63 views

vector field question

Consider the vector field $$F(x,y,z)=(zy+\sin x, zx-2y, yx-z)$$ (a) Is there a scalar field $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ whose gradient is $F$? (b) Compute $\int _C F\cdot dr \neq 0$ where ...
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1answer
41 views

the position vector $x(t_0)$ is orthogonal to the velocity vector $x'(t_0)$ if $x(t_0)$ is the point on the image of $x$ closest to the origin .

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

cylindrical and spherical coordinates

This is a very hard question to explain. In vector analysis, when dealing with surfaces, stokes theorem, gauss div theorem, etc. The cylindrical coordinates are: $x=r\cos\theta$ $ $ y=r\sin\theta$ ...
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1answer
36 views

Law of Cosines, Trigonometric Angle Addition Theorems, and Dot Product Relations

Just as the derivative, slope, and gradient are essentially the same thing I've realized that the Law of Cosines, trigonometric angle addition, and dot product are saying the same thing. My question ...
2
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2answers
76 views

Intuition of Greens Theorem in the plane

I'm trying to understand a special case of Greens Theorem. Let $V: \Omega \to \mathbb{R}^2$ be a $C^1$ vector field defined an open set $\Omega \subseteq \mathbb{R}^2$. Let $\gamma$ be a ...