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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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
44 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
54 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
39 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
29 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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31 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
58 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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72 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 ...
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1answer
151 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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24 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
19 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
35 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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49 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 \}$ ...
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1answer
38 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
41 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
33 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
53 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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22 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 ...
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2answers
94 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 ...
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1answer
25 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
26 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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49 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
47 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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1answer
40 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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26 views

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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2answers
25 views

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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44 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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1answer
43 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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2answers
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
60 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
40 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 ...
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Projecting spherical components of a variable point to the unit vector of a fixed point

Consider the following vector function in spherical coordinates: $\mathbf{v} (r, \theta, \phi) = V_{\phi} (r, \theta, \phi) \mathbf{a}_{\phi} = A \delta(r - k) \displaystyle \delta \left( \theta - ...
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1answer
32 views

Basic line integral

Let $C$ be curve along surfaces $z=\ln(1+x)$ and $y=x$ from $(0,0,0)$ to $(1,1,\ln(2))$. Calculate the work done by vector field $$\mathbf{F}(x,y,z)=(2x\sin(\pi y)-e^z)\mathbf{i}+(\pi x^2 \cos (\pi ...
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1answer
31 views

Understanding Conservative and Curl

There are several things I need to clarify on Curl. 1) Is the conservativeness of a gradient field only applicable for a Closed curve? If the field is gradient and if c (curve) is not closed then ...
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1answer
63 views

find flux,using Cartesian and spherical coordinates

Find the flux of the vector field $\overrightarrow{F}=-y \hat{i}+ x \hat{j}$ of the surface that consists of the first octant of the sphere $x^2+y^2+z^2=a^2(x,y,z \geq 0).$ Using the Cartesian ...
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1answer
37 views

an iterated integral question

This iterated integral is proving harder than I thought. Evaluate by reversing the order of integration: $$ \int_{0}^{1}\left(\int_{y=x}^{\sqrt{x}}\frac{\sin y}{y}dy\right)dx $$
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Determine if the vector field $\overrightarrow{F}$ is conservative or not.

Determine if the vector field $\overrightarrow{F}=y\hat{i}+(x+z)\hat{j}-y\hat{k}$ is conservative or not. The vector field $\overrightarrow{F}=M\hat{i}+N\hat{j}+P\hat{k}$ is conservative if ...
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9 views

Cross Product of Covectors

Is the vector/cross product defined for covectors (in the dual space) or is it, strictly speaking, only defined for vectors themselves? I would imagine that it works fine for covectors but I wanted to ...
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Finding a vector potential, i.e. given $\vec{A}$, how to find $\vec{B}$ s.t. $\vec{A} = \nabla \times \vec{B}$?

I understand that this might not be unique, but is there a (relatively) painless way to generate such a 'vector potential', so for a given field $\vec{A}$, a new field $\vec{B}$ which satifies: ...
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A method of calculation coordinates in order to implement it to a code language!

lets say that we have three points A(xa,ya,za), B(xb,yb,zv), C(xc,yc,zc) with known coordinates in 3d space. Is there a method to calculate the coordinates (x,y,z) of another point D for which the ...
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1answer
17 views

Need help proving second partial derivative equation.

I need help doing a Midterm Practice Question. Attempt: I tried finding each individual partial derivative and adding them together initially. I ended up getting something extremely complicated ...
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1answer
32 views

Spread out field lines and divergence

It can be shown by calculation of divergence of a field like $(\frac{x}{\sqrt{x^2 + y^2}}, \frac{y}{\sqrt{x^2 + y^2}})$ that it's divergence is positive. But I can't understand the geometrical essence ...
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21 views

Integrating a vector field, not uniquely defined in which sense?

I'm trying to complete an exercise in vectorial calculus, integrating the following vector field in the sense finding (one of) the function it derives from by taking the gradient: $$ \vec{v} = ...
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1answer
54 views

Use the Stokes' Theorem to find the work of the vector field $ \overrightarrow{F}$

I have the following exercise: "Use the Stokes' Theorem to find the work of the vector field $ \overrightarrow{F}=x^2 \hat{i}+2x \hat{j}+z^2\hat{k}$ along the anti-counterclockwise oriented area of ...
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2answers
27 views

Question on vector fields

Which ones are vector fields? (I checked my answers) Temperature of room at given point The gravitation that object with mass creates (x) The density of an object at given point Function $f: ...
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1answer
24 views

domain using level curves?

Let $f(x,y)=4x^2-y^2$. The problem is to determine the range of this function using the idea of level curves. So one sets $f(x,y)=D$, where $D\in\mathbb{R}$. What values can $D$ take?
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1answer
45 views

How to prove sum of vectors with same magnitude is equal to zero.

Suppose that we have $n$ vectors $v_1,v_2...v_n$ with same magnitude in plane s.t. the angle between $v_i$ and $v_{i+1}$ is $2\pi/n$ then $v_1+v_2+...v_n=0$ for all $n \geq 2$. I can show this by ...