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
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Vectors with given angle and magnitude

Give an example of vectors $\mathbf{v}$ and $\mathbf{w}$ such that the angle between $\mathbf{v}$ and $\mathbf{w}$ is $\frac{2\pi}{3}$ and $\|\mathbf{v} \text{ x } \mathbf{w}\|=\sqrt{3}$. Should I ...
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1answer
37 views

Verification of the Stokes theorem for the surface that is a part of a cone

In the solution to this question in order to work out the surface integral you can project on to $z=3$ and evaluate over the region $x^2+y^2\leq9$. I was just wondering whether or not you could do ...
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1answer
23 views

Matrices as sets of vectors

What exactly does it mean when someone says a matrix may be intrepreted as a set of vectors? As in: "A matrix can be considered a set of vectors, organised as rows or columns" It seems it would only ...
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1answer
30 views

Finding the area of a triangle from vertices? Linear Algebra

I pretty much did this problem, but I failed to get the few last blanks where they ask the area. Its confusing, they say its half the volume of matrix (u v w) in the start of the question. which means ...
3
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2answers
37 views

how to prove gradients vectors are the same in polar and cartesian coordinates.

Suppose $T=T(r,\theta)=G(x,y)$ How do you prove $\nabla T(r,\theta)=\nabla G(x,y)$? I can think of some arguments in favor of this equality, but I want an actual proof or a very good intuitive ...
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1answer
17 views

Areas of tetrahedron surfaces - how to calculate?

Reading up on Cauchy's stress theorem, I have stumbled over the so-called Cauchy tetrahedron, which is an important part of the theorem's proof. The following is cited straight from Wikipedia, but a ...
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2answers
34 views

Show that there is no vector field with curl $x \hat i + y \hat j + z \hat k$.

I have no idea how to prove this. By assuming the field has the curl I get these 3 equations: $$x = \frac{\partial F_{3}}{\partial y} - \frac{\partial F_{2}}{\partial z}$$ $$y = \frac{\partial ...
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1answer
17 views

What is the difference between single and double modulus signs. Do they both mean magnitude?

What's the difference between a set of single modulus and a set of double modulus signs? On textbooks I have seen the magnitude of two vectors vector as |x-y| but I've seen other sites where they've ...
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2answers
34 views

derivatives of a vector of functions with respect to a vector

Let $\vec W \in \mathbb R^3$. What is the general solution to: $$\frac{\partial}{\partial \vec{W}} \begin{pmatrix} f(\vec W) \\ g(\vec W) \end{pmatrix} $$ I think that in the ...
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2answers
40 views

Is $\nabla \cdot (\mathbf{a} \mathbf{b}^\mathrm{T}) = \mathbf{b}(\nabla \cdot \mathbf{a})+(\mathbf{a}\cdot \nabla) \mathbf{b}$ correct?

Wikipedia says that the following statement is a vector identity: $$\nabla \cdot (\mathbf{a} \mathbf{b}^\mathrm{T}) = \mathbf{b}(\nabla \cdot \mathbf{a})+(\mathbf{a}\cdot \nabla) \mathbf{b}$$ Where ...
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1answer
30 views

Levi-Cevita symbols: Why is $\epsilon_{ijk}\epsilon_{pjk}$ equal to $2\delta_{ip}$, but not $0$?

I'm learning vector calculus on my own and sometimes strange things happen that I don't know how I should explain them. We have this famous equality: ...
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0answers
18 views

Vanishes of second order

I encountered the following statement in an article: Let $f(X) = f(X_1, ..., X_N) = \sum_{|\alpha| \geq 2} {a_{\alpha}X^{\alpha}}$ be a convergent power series which vanishes of second order at ...
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1answer
24 views

Rewriting integrals over spheres involving $1/|x|$

The following derivation cames from calculations related to the Laplace equation and its fundamental solution. Let $g(x)$ be a test-function (meaning compact support and infinitely differentiable), ...
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2answers
80 views

In three dimensions, the Laplacian of $1/r$ is $0$ outside the origin

Why does the following hold? $$\Delta_{3}\frac{1}{r}\Bigg\vert_{\mathbb{R}^3 \setminus \left\lbrace 0 \right\rbrace}=0$$
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0answers
59 views

Gradient is covariant or contravariant?

I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis i , j , k so by invariance nature of vectors, component of gradient ...
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0answers
38 views

find the angles of a given vector sum

Assume you have n vectors in 2D space, with different fixed magnitudes $l_i$. The problem is to find the angle of each vector such that vector sum is a specific vector. That is, $\sum l_i \cos ...
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0answers
31 views

Surface Integrals, orientation and parametrizations.

I'm trying to solve the following problem: Integrate $f(x,y,z)=(x,y,z)$ over the surface $z=12$ $x^2 + y^2 \leq 25$ I parametrized the surface with $\sigma (r, \theta) = r \sin(\theta), r ...
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2answers
48 views

Requirement for continuity of unit normal vector

When considering a subset $\Omega \subset \mathbb{R}^{n}$. If we consider $\nu$, the outward unit surface normal to $\partial \Omega$, what are the requirements of $\partial \Omega$ which will ...
0
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2answers
41 views

vectors in a space (very simple question)

My question is too stupid to be googled, so I'll ask it here (because I didn't get any answers from google). Context: I have a three dimensional space and the units for x,y,z are given in meters. On ...
2
votes
1answer
23 views

Taylor expansion of the electrostatic potential $1/\|\cdot \|$

I have stumbled over this problem several times in electrodynamics, and I just don't get the hang of it. The task is to do a Taylor expansion of $\,f(\vec{x},\,\vec{a}) = ...
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1answer
32 views

Compute gradient of this expression more quickly

I want to compute $$\vec{n}\cdot \nabla^\prime G(\,\vec{r},\,\vec{r}^\prime)$$ with $$G(\,\vec{r},\,\vec{r}^\prime) = ...
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0answers
30 views

Just a curious question. What type of mathematics do we need to deal with a vector valued- function that depends on multiple vectors?

We have a scalar valued function that depends on multiple variables (takes in a vector), what about a vector valued function that depends on multiple vectors??
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0answers
51 views

How do you solve this differential equation? $\tfrac{dx}{dz} = i (M x)$

How do you solve this differential equation : $\tfrac{dx}{ dz} = i (M x)$ where $M$ is a tridiagonal matrix with elements $100$. That is, $M$ is an array with $100$ elements in triagonal form, ...
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1answer
32 views

How to denote matrix when writing equation for exam

I'm reading an article on Kalman filter on the web: http://www.cs.cmu.edu/~motionplanning/papers/sbp_papers/integrated3/kleeman_kalman_basics.pdf Something that I noticed is that bold text is used ...
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2answers
30 views

vector question assistance

let there be 2 lines: $(2,-3,1) + s(3,-2,1)$ and $(2,-1,-3) +t(3,-2,1)$ which are parallel to each other. find the formula of the plane determined by them. my try: a vector perpendicular to ...
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4answers
46 views

algebraic representation of a line in 3d

Is an algebraic representation of a line in 3d possible, or there can be only a parametric one?
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2answers
287 views

Boundary under transformation of a closed curve from $R^2\to R^3$

Consider some mapping $\phi: R_{uv} \to S\subset \mathbb{R}^3$ where $R_{uv}\subset \mathbb{R}^2$ and such that it is a simply connected region. We call the boundary of the surface (which we ...
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0answers
14 views

Deriving lower bound for eigenvalues of laplace operator

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 \quad \mbox{ on } \quad D, \quad u = 0 \quad \mbox{ on } \partial D $$ and let $w$ be a function such that $\Delta w + \beta w < 0$ ...
2
votes
0answers
70 views

Show that $L[v^2] := \Delta(v^2) + \frac{2}{w} \sum_{i=1}^n w_{x_i} (v^2)_{x_i} \ge 0$

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 $$ for some $\lambda \in \mathbb R$, also let $w$ be a function such that $$ \Delta w + \beta w < 0. $$ for some $\beta \in \mathbb R$. ...
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2answers
26 views

Question on derivation of vector identites and using some symbolic manipulations

Let $f,g : \mathbb R^n \to \mathbb R$, then for the gradient we have the product rule $$ \nabla(fg) = (\nabla f) \cdot g + f \cdot (\nabla g). $$ And by $\Delta(f) = \mbox{div}(\nabla(f)) = \nabla ...
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1answer
48 views

evaluating surface integral with divergence theory

If I have to calculate the surface integral of $\iint_S A \cdot n\ \mathrm {ds}$ where $A= 3zi-2xj+5x^2zk$ and $S$ is the surface of the cylinder $x^2+y^2=4$ and lying between $z=0$ and $z=4$ in the ...
0
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1answer
38 views

Arc Length with Vector-Valued Functions, Part B

Consider the path of a particle in a conservative force field represented by the vector-valued function $$r(t)= \left(4(\sin t−t \cos t), 4( \sin t+t \sin t), \frac{3}{2} t^2 \right).$$ A) Find the ...
0
votes
1answer
20 views

Surface integral using Stokes' theorem

$$ \int_\Gamma y\,dx+z\,dy+x\,dz $$ when $\Gamma$ $= \{ (x,y,z): x^2+y^2+z^2=9\}$ $\cap$ $x+y+z=0$ There's a theorem that states: $\int_S(\nabla \times \vec F)\cdot d \vec S$= $\int_S(\nabla \times ...
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1answer
23 views

Help with vectorial analysis problem

Let $\psi : \mathbb{R}^n \to \mathbb{R}^n$ and $f: \mathbb{R}^n \to \mathbb{R}$ be continuously differentiable functions. Let $X \in \mathbb{R}^n$, $Y=\psi(X)$ and $g=f \circ \psi$. Show that $Z ...
2
votes
1answer
44 views

Vector calculation question

the points a b c d are concordantly ( 1,2,-3) , (-1,2,1) , ( 0,1,-2) , ( 2,-1,1) find formula of the plane going thorugh d and which is pararlel to plane abc calculate the volume of pyramid abcd. ...
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0answers
21 views

Vector pyramid question

Suppose we have a pyramid with two vectors known, as well as the angle between them...if we mutpliy their size by each other mutiply by the sine of the said angle and then by sixth, will we get the ...
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0answers
24 views

Jacobian of a Vector in Maple

I am trying to derive the Jacobian matrix of a vector q, which is a very big 3-element column vector including time dependent expressions wrt another 3-element vector eta. eta := Vector([r(t), p(t), ...
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1answer
55 views

Calculate area enclosed by curve

Calculate the area of the bounded surface enclosed by the curve $(x+y)^4 = x^2y$ with the help of the coordinate transformation $x = r\cos^2 t, y = r\sin^2 t$. As I see it the area is unbounded, so ...
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vote
1answer
28 views

Verification of Stokes Theorem

I want to verify Stokes Theorem for the surface $$ \Phi = \{ (x,y,z) \in \mathbb R^3 : z = x^2 - y^2, x^2 + y^2 \le 1 \} $$ and the vector field $F(x,y,z) := (y,z,x)$. For this I use the ...
8
votes
1answer
63 views

Green's identity contradicts Helmholtz theorem

Let F be a vector field on a bounded domain $V \subset \mathbb{R}^3$, which is twice differentiable, let $S := \partial V$. According to the Helmholtz Theorem, F can be decomposed, such that: $$ ...
0
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1answer
40 views

Use divergence theorem to find $\iint_S (2x+2y+z^2) dS$ Where $S$ is the sphere $ x^2+y^2+z^2 = 1$

I tried a lot but it gets ugly really soon, any help will be greatly appreciated. T hanks
3
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1answer
59 views

Interesting dilemma, answer not matching with stewart, My work is Included

Question : Compute flux through the upper hemisphere of $x^2+y^2+z^2 = 1$ . Where $$\textbf{F} = \left( z^2x\right)\textbf{ i }+\left[\dfrac{1}{3}y^3+ \tan z\right]\textbf{ j } + \left(x^2z+y^2 ...
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0answers
8 views

How to check if vector field is monotone?

Is there any practical way of checking if given vector field is monotone? I have definition, that $a:\mathbb{R}^n\to\mathbb{R}^n$ is monotone, if for any $x,y\in\mathbb{R}^n$ $$(a(x)-a(y))\cdot ...
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votes
3answers
39 views

Find a specific vector equation of a line that divides a angle in half.

I've been studying a little geometry on my own, and I just recently stumbled on this problem, that I'm unable to answer: Given the points A=(2,-1), B=(5,4) and C=(-7,8), find a vector equation of a ...
0
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1answer
28 views

Using Gauss's Theorem in weak formulation

Can anyone see how exactly Gauss's theorem used in the following case: In defining the weak solution to the linear elliptic equation we start with $$-\sum_{j,k=1}^{n}D_{j}(a_{jk}D_{k}u) + cu = f ...
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0answers
20 views

Identity proof vector calculus

I encounter massive problems tackling the following math problem. Especially the notation in line #2 with the $\frac{\partial u}{\partial x^2}$ and in line #3 the $ U:]0,R[\times\mathbb R\ $ , ...
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1answer
46 views

A question about vector fields and divergence

I am reading the paper http://www.goshen.edu/physix/mathphys/gco/TensorGuideAJP.pdf in order to gain a basic understanding about tensors. I had some difficulties about understanding some definitions. ...
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2answers
41 views

is it necessary that curl of 2d vector is perpendicular to the plane.

I am just confused, help me guys. The question comes up, because we say that curl is either clockwise or anti-clockwise at a point.
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2answers
68 views

Integral and area in a plane [closed]

Show that the value of the following integral is proportional to the area included in a curve $C$: $$\oint_C3y\,dx+3z\,dy-x\, dz,$$ where $C$ is a smooth, closed curve upon the $2x+2y+z=2$ plain.
1
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1answer
95 views

Vector function tough question

If a curve has the property that the position vector $\vec{r}(t)$ is always perpendicular to the tangent vector $\vec{r'}(t)$, how can I show that the curve lies on a sphere with center the origin? ...