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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Orthogonal Position and Velocity Vectors

Is it true that if the position and velocity vectors of a moving particle are always perpendicular the path of the particle is on a sphere? If so how do I prove it? Geometrically I believe it makes ...
2
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1answer
22 views

System of vector equations (in Minkowski space)

I wonder whether there is a systematic approach to find (or at least whether there are criteria for the existence of) vectors $P_0, P_1, \dots, P_n$, say in $n$-dimensional Minkowski space of ...
2
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2answers
60 views

Vector Functions of One Variable

Question A particle moves along the curve of the intersection of the cylinders $y=-x^2$ and $z=x^2$ in the direction in which $x$ increases. (All distances are in cm.) At the instant when the ...
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0answers
26 views

How to derive the error when approximating divergence using the Gauss divergence theorem?

I am trying to derive the error for approximatively computing the divergence of a vector field $\mathbf{a}$. The Gauss divergence theorem states $\int_V \nabla \cdot \mathbf{a} dx = \oint_{\partial ...
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0answers
7 views

Finding a vector, using tetrahedron and it's angles with axis

Vectors $\vec{m}$=(1,1,1) and $\vec{n}$=(2,1,1) are given. Find $\vec{p}$=(a,b,c) if it forms an angle of $\pi/4$ with Ox axis, $\pi/3$ with Oy axis, and an obtuse angle with Oz. vectors $\vec{m}$ ...
2
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1answer
21 views

Einstein summation convention: Del operator and dot product

Now, I am aware of the summation convention for the dot product $$\mathbf{a} \cdot \mathbf{b} = a_i b_i$$ But I am unsure about how to represent $(\nabla \cdot \mathbf{a}) \mathbf{b}$ and ...
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2answers
16 views

Finding vector form of an angle bisector in a triangle

Find vector form of angle bisector, $\vec{BP}$, using $\vec{b}$ and $\vec{c}$. That's how far I've got. Please don't use $tb+ (1-t)b$, or similar since I don't know what that is. Just basic dot ...
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1answer
29 views

Vector valued integral in spherical coordinates

Whenever I have been presented with integrals as (*), I have always used some sort of symmetry to get around actually calculating the integral. Now it just struck me that I have no idea how to ...
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2answers
27 views

Question about basis of Vector Space [closed]

Show that $B$ is a basis for $\mathbb{R}^2$:Where $$B=\{(-1,1) , (2,3)\}$$
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1answer
40 views

Triangle, vectors, hard to explain

P is the middle of a median line from vertex A, of ABC triangle. If Q is the point of intersection of lines AC and BP. Find relations of $|\vec{AQ}|$/$|\vec{QC}|$ and $|\vec{BP}|$/$|\vec{PQ}|$ Any ...
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1answer
21 views

Define vector using other 2 coplanar vectors

Vectors $\vec{b}$ and $\vec{c}$ are given. ∠(b,c)=2pi/3. Find vector $\vec{a}$, coplanar with $\vec{b}$ and $\vec{c}$, length $|\vec{a}|=4$ and ∠(a,b)=pi/6 I know it's something with triple product. ...
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1answer
40 views

Angle bisector between two vectors, which are expressed by unit non-orthogonal vectors

Given vectors $\,a=2m-2n\,$ and $\,b=3m+6n\,$, where $\ \left\lvert m \right \rvert =\left\lvert n \right \rvert =1\,$ and $\,\angle\left(m,n\right)=\dfrac{2\pi}{3},\,$ find vector of angle bisector ...
3
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0answers
41 views

Orthogonal curvilinear coordinates (derivatives of unit vectors)

Suppose that $\{u_i\}_{1\le i\le 3}$ is a set of orthogonal curvilinear coordinates with unit vectors $\{\mathbf{\hat{e}_i}\}_{1\le i\le 3}$. I proved that $$\frac{\partial ...
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0answers
30 views

How to use Stokes Theorem here?

I think we have to use the Stoke's Theorem here. So let $F=(-y^3+xz)i+(yz+x^3)j+(z^2)k$. Then Curl $F=-yi+xj+3(x^2+y^2)k$. Now $\int \int_S Curl F.n dS=\int_C F.dr=$The integral we have to compute. ...
1
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1answer
28 views

Some confusion about where vectors emanate from,

Take the plane x+y+z=0, for example. Then the vectors (0,0,0), (1,-1,0), (1,0,-1) "lie on this plane." And to find a normal to this plane, just compute the cross-product of any two independent ...
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1answer
34 views

What is nds in the surface integral of Stokes' Theorem?

I am working on an integration problem, where Stokes' theorem seems to be applicable. I have found the curl of F, and the normal vector, n, to the surface. The surface that I will integrate over is ...
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1answer
17 views

volume of the parallepiped spanned by the vectors

Hi I am having difficulty with part (2) of the following proposition. Suppose that $x,y,z\in\mathbb{R}^3$, then (1) $\|x\times y\|=\|x\|\|y\|\sin\theta$ is the area of the parallelogram spanned by ...
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2answers
48 views

How to determine if division by zero causes a derivative to not exist?

I study engineering so I don't know calculus with sufficient rigor to know if a derivative exists or not. If one wants to take the divergence of the field $\vec{E}=\frac{\hat{r}}{r^2}$ the typical ...
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1answer
61 views

Mean value theorem for vector valued function (not integral form)

Let $f:U\to\mathbb R^m$ be differentiable with $U\subseteq \mathbb R^n$ being open and convex. If $f$ is absolutely continuous, then by fundamental theorem of calculus we have following version of ...
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3answers
180 views

Intuitive explanation of div(curlF)=0 [duplicate]

If we consider $\mathbf{F}$ as a vector field, then we say that $\mathrm{div}(\mathrm{curl}(\mathbf{F}))=0$. We can prove this in mathematics easily. But I' am not getting an intuitive explanation due ...
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3answers
45 views

Evaluate the surface integral for a vector field

I asked this question among others in another thread (Vector analysis questions). I was given a link to a site (http://mathinsight.org/surface_integral_vector_field_introduction) which I read, and ...
1
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1answer
26 views

Equations of Motion in Polar Basis

A particle of mass m moves under a central force field $ \mathbf{F}=-k\mathbf{r}$ where k is a constant with dimensions $ N m^{-1} $. Assuming that the particle moves in the equatorial plane ( ...
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0answers
30 views

Connection between covariant and contravariant components o tensor

What is the general proof of the relation between covariant and contravariant components of a tensor using the metric tensor? $${g^{mr}g_{rn}=\delta^{m}_{n}}$$
2
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4answers
58 views

Showing that gravitational flux remains constant.

Let the vector field $$\vec{F}(x,y,z)=\frac{GM}{(x^2+y^2+z^2)^\frac32} \begin{pmatrix} x \\ y\\ z\\ \end{pmatrix}$$ Where $G$ is the universal gravitational constant and $M$ the mass of earth. I ...
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1answer
37 views

Vector analysis questions

I need some help with the following questions, I have made an attempt at both which is below but don't think its right. Any help in where I'm going wrong would be appreciated. 1) $$\int_{0}^{2} ...
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1answer
19 views

Analysing data to find the principal parameter

I have a data set $N\times M$, which contains information about motorcycles: $N$ motorcycles have been sold during some time and for each bike there is $M$ parameters regarding the sale such as price, ...
0
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1answer
35 views

Prove $\unicode{x222F}_S{f(x,y,z)d\vec{S}}=\iiint_E{}\nabla f(x,y,z)dV$.

Let the closed surface $S$ be delimit volume $E$ and let the scalar function $f$. I need to demonstrate that : $$\unicode{x222F}_S{f(x,y,z)d\vec{S}}=\iiint_E{}\nabla f(x,y,z)dV$$ I do have some ...
3
votes
1answer
47 views

How to use the $b\cdot\nabla$ operator?

While trying to prove $$[c\cdot (b\cdot\nabla) - b\cdot(c\cdot\nabla)]a = (\nabla\times a) \cdot (b\times c)$$ I had some difficulties on how to treat the term $(b\cdot\nabla)$. It seems that ...
3
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3answers
54 views

Interpretation of the curl of a vector field

Let us assume the curl of a vector field is $$ P=(xy)(a_x)+ (y z) (a_y) +(z x) (a_z) $$ Where $ a_x, a_y, a_z $ are unit vectors along x y and z . Then is the curl at a point in the field the ...
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3answers
97 views

Showing that $\nabla\times(\nabla\times\vec{A}) = \nabla(\nabla\cdot\vec{A})-\Delta\vec{A}$

I have problems to demonstrate: $\nabla\times(\nabla\times\vec{A}) = \nabla(\nabla\cdot\vec{A})-\Delta\vec{A}$. I don't have any clue how can I start to work with it. Any hint will be helpful.
2
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2answers
46 views

Curl of a vector field cross itself

How we can use the property that $$A×(B×C) = B(A.C)- C(A.B)$$ to prove the relation: $$a×(∇×a) = ∇ (a^2/2) -(a.∇)a.$$ When I use it, the result directly appear to be $$∇(|a|^2 )-(a.∇)a$$ instead of ...
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3answers
91 views

How to differentiate the following interesting vector product?

How do we differentiate the following vector product with respect to $\boldsymbol r$. \begin{equation} \frac{d}{d\boldsymbol r}\bigg[(\boldsymbol \omega \times\boldsymbol r)\cdot (\boldsymbol \omega ...
2
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0answers
32 views

Vector/Tensor analysis, Elastic Waves

So I'm fairly confused at the moment. For reference, I'm reading this document, and the current area of interest is Section 7: Characteristic Surfaces for Planar Waves. I'm not gonna give too much ...
3
votes
2answers
85 views

Stokes Theorem. Where is my mistake?

Use Stoke's Theorem to prove that the following line integral has the indicated value. $$ \int_\mathscr{C} y \,dx +z\,dy+x\,dz = \pi a^2 \sqrt{3}$$ where $\mathscr{C}$ is the intersection curve ...
4
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1answer
55 views

Choice of order in the Leibniz rule is arbitrary?

One of the rules which characterizes the exterior derivative is that, for $\varphi$ a real-valued function and $\omega$ a $k$-form, we have $$d(\varphi \cdot \omega) = d\varphi \wedge \omega + ...
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1answer
63 views

How does gradient of a vector point steepest ascent

The derivative of distance function with respect to time give velocity function in single variable calculus. But how does gradient of a multivariable function point steepest ascent? I have been ...
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0answers
14 views

Derivative of product of Vector Transpose and Vector

I was reading about simulation of cloth in graphics where I found this part a little difficult to understand : Firstly, from what I understand, he considers a force C(x) ...
2
votes
1answer
20 views

Algebra with differential operators (Alternative forms of the Laplacian in spherical coordinates)

Given is the following: $$\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \,\frac{\partial f}{\partial r} \right) = \frac{\partial^2 f}{\partial r^2} + \frac{2}{r} \frac{\partial f}{\partial ...
1
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1answer
12 views

How to divide a vector on a sphere into northern and southern components?

Suppose we have $S^2$ and a vector $\vec{A}$ pointing at a random direction. Let us divide the sphere into $S_N$ for $0 \leq \theta \leq \frac{\pi}{2}$ and $S_S$ for $\frac{\pi}{2} \leq \theta \leq ...
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0answers
25 views

Can you prove that the integral below, with a vectorial field, is zero?

If $\vec{J}(\vec{r})$ is a vector field limited in infinity. Prove that the integral below is zero: \begin{equation} ...
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0answers
32 views

Coordinate Systems Transformation(Rectangular to Cylindrical)

I am new to this subject: Cartesian, Cylindrical and Spherical coordinate system. Coordinate System Transformation I have this example problem that I cant get the right answer. Transform to ...
1
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2answers
22 views

Volume inside loop using Green's theorem.

Let $\mathcal{C}$ be the curve defined by the vector function $\vec r(t)=(1-t^2)\vec i+(t-t^3)\vec j$ with $t\in \Bbb R$. I need to find the area confined in the closed loop $\gamma$ formed by ...
2
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2answers
25 views

Line integral of 3 segments, Green not applicable…

Let $\mathcal{C}$ be the 3 segments successively going from $(0,0,0)$ to $(2,4,6)$ to $(3,6,2)$ and to $(0,0,1)$. I need to calculate the work made by the vector field : ...
6
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0answers
57 views

basic question from vector analysis

Let $\vec v = (a,b,c)$ be a constant vector and let $\vec r = (x,y,z)$ denote the position vector. Consider the vector field $\vec v \times \vec r$. A straightforward calculation (using determinants ...
1
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1answer
34 views

How to prove this identity in vector calculus (suffix notation)?

Let $\epsilon_{ijk}$ be the alternating tensor defined by $$\epsilon_{ijk} = \begin{cases} 0, & \text{if any of $i$, $j$, $k$ are equal}\\ 1, & \text{if $(i,j,k)=(1,2,3)$, $(2,3,1)$ or ...
0
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1answer
46 views

Components of a vector product as an antisymmetrical rank 2 tensor

So I'm reading Landau and Lifshitz' Theory of Elasticity (https://archive.org/details/TheoryOfElasticity) and they have done, among (many) other things, something I simply don't understand. On page ...
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3answers
39 views

Vector analysis : following given trajectory, will particles collide?

Let two particles move by a trajectory respectively given by $\vec{r_1}(t)=t\vec{i}+t^2\vec{j}+t^3\vec{k}$ and $\vec{r_2}(t)=(1+2t)\vec{i}+(1+6t)\vec{j}+(1+14t)\vec{k}$. In my vector analysis course, ...
2
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2answers
35 views

Proving that $\vec F=yz(2x+y+z)\hat i+zx(x+2y+z)\hat j+xy(x+y+2z)\hat k$ is conservative field

I need to prove that $\vec F$ is conservative field $$\vec F=yz(2x+y+z)\hat i+zx(x+2y+z)\hat j+xy(x+y+2z)\hat k$$ My attempt: $\vec{F}$ is conservative iff $\nabla \times \vec{F} = 0$ $$ ...
1
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2answers
38 views

Proving that $\vec F$ is conservative field

I need to prove that $\vec F$ is coservative field: $$\vec F=\underbrace{\bigg(yz+\frac{1}{yz} \bigg)}_{Q} \hat i+\underbrace{\bigg(xz-\frac{x}{y^2z} \bigg)}_{P}\hat ...
0
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1answer
43 views

Integral of divergence equal to divergence of integral?

Just as the heading reads...is the integral of the divergence of a vector field equal to the divergence of the integral of a vector field? $\int\nabla\cdot\vec U dz = 0$ same as ...