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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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 ...
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17 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?
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29 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 ...
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3 views

Gentle introduction to discrete vector field [on hold]

I am looking for a gentle introduction to discrete vector field. Thanks in advance.
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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 ...
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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 ...
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1answer
23 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 ...
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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:$$ ...
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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.
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1answer
60 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 ...
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27 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 ...
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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 ...
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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 ...
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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 $\int_C fds$, ...
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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 ...
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1answer
19 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 ...
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20 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 ...
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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 ...
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1answer
47 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 ...
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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 ...
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1answer
19 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 ...
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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 ...
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1answer
52 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: $$z=g(x,y)=-x-y$$ ...
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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 ...
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128 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 ...
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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.
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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 ...
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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 ...
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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.
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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 \}$ ...
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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 ...
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1answer
37 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: $$\nabla (\mathbf a \cdot \bf r)=\mathbf a $$
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1answer
28 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 ...
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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 ...
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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 ...
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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 ...
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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} ...
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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} ...
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24 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 ...
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1answer
42 views

$\int_{-c}\mathbf{F}.\mathbf{dl}=-\int_{c}\mathbf{F}.\mathbf{dl}$-what is wrong here?

We know about line integral that $\int_{-c}\mathbf{F}.\mathbf{dl}=-\int_{c}\mathbf{F}.\mathbf{dl}$. Suppose my $\mathbf{F}$ is $\frac{\mathbf{r}}{r^3}$ and path is radial path from $r=a$ to $r=b$. so ...
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37 views

Help with surface integral question

Find the surface area of the plane $$x+\frac{1}{\sqrt{2}}y+\frac 14 z=1$$ limited by the coordinate system planes My findings : I suppose we should express the scalar $z=f(x,y) \rightarrow ...
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normal of a function

Function $$\mathbf{r}(u,v)=(2u+2v)\mathbf{i}+(-3+v^2)\mathbf{j}+(2u^2)\mathbf{k}$$ is a parametrization of a surface. What's the normal vector of this surface at the point $(u,v)=(1,-2)$. What I got: ...
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Helmholtz decomposition - motivation

Our lecturer presented us the Helmholtz decomposition of smooth vector fields. He added a proof, but he didn't provide any single motivation - e.g. where Helmholtz used the decomposition or for which ...
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42 views

Imaginary part of Log laplacian

I'm confused about how to calculate $\nabla^2 \log z$, where $z=re^{i\theta}$ is a complex number. My calculations return $$ \nabla^2 \log z = 2\pi\frac{\delta(r)}{r} [\delta(\theta) + i ...
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$\operatorname{div}$ and $\operatorname{grad}$ in spherical coordinates. Formula from general relativity goes crazy

I try to calculate the gradient of a function and the divergence of a vector field in spherical coordinates. Nothing special so far, but a formula that I learned in a general relativity lecture ...
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34 views

Vectors Grade 12 Problem

$1a)$ $a$ and $b$ are position vectors of points A and B on the plane. Show that the line that passes through A and B has the vector equation $r= sa - (s-1)b$ $1b)$ What value of s does the point P ...
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38 views

Need a little help with these vectors

Let $v_1$, $v_2$, $v_3$ be mutually orthogonal non-zero vectors in $3$-space. So, any vector $v$ can be expressed as $v = c_1 v_1 + c_2 v_2 + c_3 v_3$. (a)Show that the scalars $c_1$, $c_2$, $c_3$ ...
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Evaluation of an Integral in Vector Analysis

I'm trying to calculate an individual probability $P(\hat{a})$ from a joint probability $P(\hat{a},\hat{b})$ in a physics application, where $\hat{a},\hat{b}$ are unit vectors. I need to evaluate the ...
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1answer
49 views

Which of these things is not like the others?

What's in a name? Well quite a lot, if you're confused enough. I have an engineering-style mathematics education, based on good old hand waving and learning bits and pieces from all over the place. I ...
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1answer
30 views

Field lines of vector field

Okey if $\phi(x,y)=\ln(x^2+y^2), (x,y) \neq (0,0)$. Find the field lines for $\mathbf{G}=\nabla \phi$. So $\mathbf{G}=\frac{2x}{x^2+y^2}\mathbf{i}+\frac{2y}{x^2+y^2}\mathbf{j}$ right? To find the ...