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
0answers
20 views

Vector Field in $\mathbb{R}^3$

Consider a collection of $n+1$ mass weighted points in $\mathbb{R}^3$. Suppose we have one mass located at the point (0,0,0) with mass $m\in\mathbb{N}$ and further suppose we have $n$ masses arranged ...
2
votes
0answers
36 views

Green's Theorem

Hey guys I am having difficulties in problem 5. I thought I understood it, but I suppose I was mistaken. I will now explain what I planned to do to solve this problem and where I got stuck. So I ...
0
votes
0answers
25 views

approximating a sphere

Suppose that $R$ is a simple connected region in $\mathbb{R}^3$, enclosing a volume $V$. I am looking at ways to approximate $V$ using spheroidal volume elements. The traditional approach is to use ...
2
votes
1answer
22 views

Surface integral is $0?$

I have a quick question: if $\mathbf{f}:\;\mathbb{R}^3\to\mathbb{R}^3$ is odd, in the sense that $\mathbf{f}(-\mathbf{v})=-\mathbf{f}(\mathbf{v})$ for any $\mathbf{v}\in\mathbb{R}^3$, and $S$ is a ...
0
votes
1answer
26 views

Stokes' Theorem and Surface Independence Failure

As we know, if $\vec{F}=\nabla\times\vec{A}$ then from Stokes' Theorem, $\iint_{S_1} \vec{F}\dot \,d\vec{S}=\iint_{S_2}\vec{F}\dot \,d\vec{S}$ where $S_1$ and $S_2$ have the same boundary. Does ...
0
votes
2answers
28 views

why are conservative vector fields curl-free?

The book told me that, if a vector field $\vec{F} = Mi + Nj$ is conservative, then $$ M_y = N_x $$ But why is this true?
0
votes
0answers
19 views

Three linearly independent vector fields

How can one find three linearly independent vector fields on $S^1\times S^2$? I know that $S^1\times S^2 \cong SO_3( \mathbb{R})$, i.e. the set of orthogonal $3 \times 3$ matrices with determinant ...
0
votes
0answers
23 views

vector field problems

I'm trying to review some problems on vector fields for the final, and would appreciate if someone can tell me whether my answers are right, so I know if I'm doing it correctly: $f$ is a vector ...
0
votes
0answers
24 views

Flux through surface with no stationary points inside it is zero?

Seems intuitive that a flux through a closed surface with no stationary points inside this surface is zero. Is it really so? And if yes can this flux be expressed by some characteristics of these ...
3
votes
1answer
23 views

Why should we expect the divergence operator to be invariant under transformations?

A lot of the time with vector calculus identities, something that seems magical at first ends up having a nice and unique proof. For the divergence operator, one can prove that it's invariant under a ...
0
votes
2answers
28 views

Find the flux of the vector field across the boundary of the cube

Find the flux of the vector $F=e^{xy} \hat{i} +e^{yz} \hat{j} +z \hat{k}$ across the boundary of $[0,1] \times [0,1] \times [0,1]$. Can someone tell me the setup of this problem?
1
vote
0answers
38 views

packing spheroids/ellipsoids

I am thinking about a problem I'm trying to formulate, from a mathematical analysis viewpoint. Suppose that you have a finite region in 3-space, and you want to populate it with spheroids or even ...
0
votes
0answers
30 views

How to convert vector field from cartesian to spherical

I have a vector field $A ( r) = \omega \times r$, where $r=(x,y,z)^T$ and now I want to express this field in cylindrical coordinates. How do I do this?
0
votes
0answers
37 views

How to find $\nabla$ in spherical coordinates

I want to derive(!) just a few components like the $\hat{e}_r$ component of the divergence operator in spherical coordinates and the $\hat{e}_{\phi}$ component of the curl operator in spherical ...
0
votes
0answers
3 views

Gentle introduction to discrete vector field [closed]

I am looking for a gentle introduction to discrete vector field. Thanks in advance.
0
votes
0answers
12 views

Showing orthogonality of coordinate surfaces are orthogonal for oblate spheroidal co-ordinates.

So oblate spheroidal co-ordinates are defined as: $$x = \cosh R \cosθ \cos φ$$ $$y = \cosh R \cosθ \sin φ$$ $$z = \sinh R \sin θ .$$ To show the coordinate surfaces for $R$, $\theta,\phi$ are ...
0
votes
0answers
14 views

Constructing a vector field with a given divergence

Let $f$ be a scalar-valued trivariate function defined on some bounded domain, say $[a_1,b_1]\times[a_2,b_2]\times[a_3,b_3]$. I would like to construct a vector field ...
0
votes
1answer
26 views

Vectors and Planes

Let there be 2 planes: $x-y+z=2, 2x-y-z=1$ Find the equation of the line of the intersection of the two planes, as well as that of another plane which goes through that line. Attempt to solve: the ...
0
votes
0answers
25 views

Vector Question Help

A plane is determined by $(x,y,z) = (1,-1,0) + t(1,-1,2)$ and point $p(1,2,3)$. find point of intersection of $(1,4,-1)+s(-6,2,-4)$ with this plane. I tried this: given the data plane equation is:$$ ...
0
votes
1answer
17 views

Computing line integral

Compute the line integral $$\int_C A \cdot dr$$where $$A = x^2 \vec{i} + y \vec{j} + (xz - y)\vec{k}$$ from $(0,0,0)$ to $(1,4,7)$ along the line segment joining the two points.
1
vote
1answer
64 views

Vector Calculus Surface Integral (Limits of Integration)

I'm currently having trouble with the following problem. I believe that I have most of the problem set up, but I am having trouble finding what the limits of integration should be. $\int\limits_S ...
0
votes
2answers
33 views

Line integral over a curve in the II quadrant

I am lost here: $C = x^2 + y^2 = 4$ from $(0,2)$ to $(-2, 0)$. Calculate $ \ \int_c y^2 ds \ \ $ and give reasons the sign is correct. It's obviously the circular arc going counterclockwise from ...
0
votes
1answer
21 views

Question about vector equations of lines and planes

Find the equation of the line going through the point $(2,-3,4)$ ,and which is perpendicular to the plane $ x+2y + 2z = 13$ So I tried this: the normal of the plane is $(1,2,2)$, random point on the ...
0
votes
2answers
43 views

Vectors Question

I have a question regarding Vectors; Find the equation of the plane perpendicular to the vector $\vec{n}\space=(2,3,6)$ and which goes through the point $ A(1,5,3)$. (A cartesian and parametric ...
0
votes
1answer
21 views

How to think about integrals along C.

One of the ways I like understanding things is being able to "see what's going on" so I can hypothesise intuitive results (and then rigorously prove them later). For example, when I see $\int_C fds$, ...
1
vote
0answers
16 views

The difference between $(\frac{\partial F}{\partial T})_X$ and $\frac{\partial F}{\partial T}$?

Just as the headline says, what is the difference between $(\frac{\partial F}{\partial T})_X$ and $\frac{\partial F}{\partial T}$ ? The former is used at least in thermodynamics, and I find the ...
1
vote
1answer
20 views

Total and partial derivatives of $F(T, X(Y, Z))$ respect to $T$?

So, yeah, haven't been to vector analysis just yet, but for now I'd need to make sure I understand the definition of total derivatives (and partial derivatives). The question is simple. I wish to ...
0
votes
0answers
21 views

Generalized functions and vector calculus theorems

To apply the divergence theorem, for example, there are conditions on your function. Your function must be a function in the ordinary sense in the first place. But in Electrodynamics, sometimes our ...
1
vote
2answers
34 views

If $\mathbf{A} \times \mathbf{B} = \mathbf{C}$, $\mathbf{A}$ and $\mathbf{B}$ are unique?

As in the title, I know that a vector $\mathbf{C}$ is obtained by two vectors $\mathbf{A}$ and $\mathbf{B}$: by hypothesis, they are both entirely lying in a plane orthogonal to $\mathbf{C}$ and they ...
1
vote
1answer
48 views

Line integral with Stokes

Let $C$ be curve $(x-1)^2 + (y-2)^2 =4$ and $z=4$ orientated counterclockwise when viewed from high on the z-axis. Let $$\mathbf{F}(x,y,z)=(z^2 +y^2 +\sin ...
1
vote
1answer
33 views

Basic surface integral with Stokes.

Calculate surface integral $\iint_S \nabla \times \mathbf{F} \bullet \mathbf{n} \; dS$ with stokes, when $$\mathbf{F}=\left\langle\frac{5y(z-1)}{6},xz, 6e^{xy}\cos{z}\right\rangle$$ and $S$ is surface ...
0
votes
1answer
20 views

Question with divergence theorem

Calculate flux of field $$\mathbf{F}=(3x^2 y+6)\mathbf{i}+\left(\frac{x y^2 +1}{3}\right)\mathbf{j}+(3yz^2+3)\mathbf{k}$$ in a box where it's opposite angles are in $(1,1,1)$ and $(2,2,2)$ and it's ...
0
votes
0answers
23 views

Question about divergence

I completely understand why divergence of a vector field means flux density, but... I've read a different interpretation of divergence, which is the expansion rate of an infinitely small ball. I kind ...
2
votes
1answer
53 views

Question with Stokes theorem

Show with Stokes that $\oint_C (y\mathbf{i}+z\mathbf{j}+x\mathbf{k})\bullet d\mathbf{r}=\sqrt{3}\pi a^2$ when $C$ is intersection of $x^2+y^2+z^2=a^2$ and $x+y+z=0$. My work: $$z=g(x,y)=-x-y$$ ...
0
votes
0answers
71 views

Calculus , integration by parts

Why does given that the 4-vector $V^\mu (x)\to 0$ as $x^\nu\to \infty $ imply that $$\int d^4x \,\,\, \partial_\mu V^\nu(\partial^\mu V_\nu-\partial_\nu V^\mu)=0$$? I tried integrating by parts. That ...
1
vote
1answer
130 views

Line integral using Green's theorem.

Let $C$ be parametrization $\mathbf{r}=\sin(t) \mathbf{i}+\sin(2t) \mathbf{j}$, $t \in [0, 2\pi]$. Sketch picture and investigate how the surface orientates. Calculate line integral $\oint_C ...
0
votes
0answers
19 views

How to find the tangential component of Velocity?

Let $v=-\nabla \phi$, where $\phi$ is the velocity potential. I am interested in to find the value of $|v_{tan}|$ which is the tangential component of velocity.
2
votes
0answers
25 views

A vector analysis question requiring multivariable calculus

The question is taken from a Vector-Analysis worksheet as an extension exercise, here is the first part: Let $\underline{v}$ be a vector field $\mathbb{R}^{2} \backslash (0,0)$ of the form ...
0
votes
1answer
16 views

With respect tho the $Oxyz$ axes, the temperature of a media is given by: $T=T_0 (1+ax+by)e^{cz}$

With respect tho the $Oxyz$ axes, the temperature of a media is given by: $$T=T_0 (1+ax+by)e^{cz}$$ Where $a, b, c$ and $T (>0)$ are constants. At the origin O, find the direction in which the ...
1
vote
1answer
31 views

Vector field with parallel field lines

If the field lines of vector field $\mathbf{F}(x,y)$ are parallel, what can be said about the divergence and curl of the field? I think the curl must be zero but divergence can get value.
1
vote
0answers
37 views

Linearization of Implicit ODE (Equations of Motion)

let's say we have a system with vector $q_{(t)}$ representing the degrees of freedom (DoF), and state vector $ x_{(t)} = \left \{ \begin{array}{c c} q_{(t)} \\ \dot{q_{(t)}} \end{array} \right \}$ ...
0
votes
1answer
22 views

Find the derivative of the scalar field: $\Omega=x^2yz+4xz^2$ at the direction of the vector: $(2, -1, -1)$ at the point: $(1, -2, -2)$

Find the derivative of the scalar field: $$\Omega=x^2yz+4xz^2$$ at the direction of the vector: $(2, -1, -1)$ at the point: $P(1, -2, -2)$ Hint: Write the unit vector $\hat n$ at the beginning, qhich ...
1
vote
1answer
38 views

If $\bf a$ is a constant vector field, and $\bf r$ is the position vector, prove that: $\nabla (\bf a\cdot \bf r)=\bf a$

If $\bf a$ is a constant vectorial field (constant magnitude and direction), and $\bf r$ is the position vector, prove that: $$\nabla (\mathbf a \cdot \bf r)=\mathbf a $$
1
vote
1answer
30 views

Vector potential question

If $\mathbf{F}$ and $\mathbf{G}$ are smooth and conservative. Find vector potential $\mathbf{H}$ for $\mathbf{F} \times \mathbf{G}$. I tried to find it like this (kinda brute force-ishly) $$\small ...
1
vote
1answer
34 views

Vector differential identities

Proof that $\nabla\bullet(f(\nabla g\times \nabla h))=\nabla f \bullet(\nabla g \times \nabla h)$. When $f,g$ and $h$ are smooth scalarfields. Can I expand $\nabla\bullet \overbrace{(f(\nabla g\times ...
0
votes
0answers
16 views

How to find a resultant vector for given multiple vectors lie on different position.

I have modelled rectangular features and then I compared my model with a reference model. (Assume everything is in 3D space) . In the below figure, Reference model is shown in light pink ...
2
votes
2answers
75 views

Apply Chain rule to vector function with chained dot and cross product?

Okay, I got $\mathbf{v}=(\mathbf{u}_{n-1}-\mathbf{u}_{n})\times(\mathbf{u}_{n}-\mathbf{u}_{n+1})$ and ...
0
votes
1answer
19 views

3D Fourier transforms of $e^{-\beta r} $ and $re^{-\beta r} $

I am trying to find the integrals $$\large\int\limits_{\mathbb{R}^3} e^{-\beta \left|\vec{r}\,\right|}e^{i \vec{q} \cdot\,\vec{r}} \mathop{d^3r}$$ $$\large\int\limits_{\mathbb{R}^3} ...
0
votes
1answer
23 views

An integral inequality with little information

$u,v$ are scalar fields on $V\subset\mathbb{R}^3$ such that $\nabla^2 u=0$ on $V$ and $u=v$ on $\partial V$. Prove that: $$\int_V|\boldsymbol{\nabla} ...
0
votes
0answers
25 views

Books which explain vector analysis/algebra in detail.

I'm trying to learn vectors but I can't find a decent book which explains vectors in depth. I need a book which explains vectors from the beginning, using a beginner's approach(assuming the reader ...