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).

learn more… | top users | synonyms

1
vote
1answer
16 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 ...
1
vote
2answers
33 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 ...
2
votes
1answer
27 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: ...
0
votes
0answers
17 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 ...
0
votes
1answer
23 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), ...
0
votes
2answers
69 views

$\Delta_{3}\frac{1}{r}\Bigg\vert_{\mathbb{R}^3 \setminus \left\lbrace 0 \right\rbrace}=0$

Why does the following hold? $$\Delta_{3}\frac{1}{r}\Bigg\vert_{\mathbb{R}^3 \setminus \left\lbrace 0 \right\rbrace}=0$$
0
votes
0answers
50 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 ...
1
vote
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 ...
0
votes
0answers
29 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 ...
2
votes
2answers
47 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
votes
2answers
40 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}) = ...
0
votes
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) = ...
0
votes
0answers
27 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??
0
votes
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, ...
0
votes
1answer
28 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 ...
0
votes
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 ...
0
votes
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?
1
vote
2answers
284 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 ...
0
votes
0answers
13 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
53 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 $\beta \in \mathbb R$. ...
0
votes
2answers
25 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 ...
0
votes
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
votes
1answer
35 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 ...
0
votes
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. ...
0
votes
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 ...
0
votes
0answers
23 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), ...
1
vote
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 ...
1
vote
1answer
27 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
votes
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
votes
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 ...
0
votes
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 ...
2
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
votes
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 ...
0
votes
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\ $ , ...
0
votes
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. ...
1
vote
2answers
38 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.
0
votes
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
vote
1answer
94 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? ...
1
vote
1answer
33 views

One dimensional integrals in Green's theorem

I am trying to understand Green's theorem, but the problem is I don't know what is the definition of the integrals in the theorem. This is the expression that one proves to hold with some assumption ...
0
votes
1answer
15 views

Get a third point (lat, lng) from two given

I have two points as follow (the distance between them is variable): I need to get a third as shown: The two first points change all the time, including the distance between them. My problem: I ...
0
votes
0answers
20 views

Question related to analysis of Dot Product

I have question related to dot product of two vectors, lets say I have two position vectors $\vec a$ and $\vec b$. And |$\vec b$| > |$\vec a$| but there exact measurement is unknown. And there is ...
0
votes
1answer
27 views

change in velocity if submarine

A submarine is travelling at 20 km/h due east. A short time later it is travelling due north at 15 km/h. Calculate the change in velocity of the submarine. My answer is 25km/hr at N53.13E
5
votes
2answers
56 views

line integral: anticlockwise parametrisation in $\mathbb R^3$

Consider $\gamma$ given by the sides of the triangle with vertices $(0,0,1)^t$, $(0,1,0)^t$ and $(1,0,0)^t$. So $\gamma$ runs through the sides of the triangle. Let $f(x,y,z)=(y,xz,x^2)$. I want to ...
2
votes
1answer
74 views

A confusing vector field differential

In my notes on theoretical mechanics, I wrote that my professor stated this vector identity: $$\mathrm{d}\mathbf{P}(\mathbf{r})=[\nabla\cdot\mathbf{P}(\mathbf{r})] \mathbf{dr} + ...
0
votes
1answer
22 views

Volume of a cone in an $n$-dimensional ball

Assume that $B$ is an $n$-dimensional ball of radius $R$ centered at the origin, i.e., $B=\{x\in\mathbb{R}^n : \|x\|\leq R\}$. Fix a point $x_0$ in $B$ and $\delta \in (0,\pi)$, and let $C$ be the ...
2
votes
1answer
33 views

Reference Request: How to Parametrize Curves and Surfaces in $\Bbb R^3$

I don't feel like I have a good grasp of how to parametrize a curve or surface. I can quickly enough verify that a given parametrization DOES correspond to a curve, and I've memorized a few of the ...