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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Elementwise normal to vector of unknowns and non-defined matrix multiplications

I wonder about statement (1). It is given that u is a column vector, A and B are constant, symmetric, square matrices of such size that the expressions on the left hand side of (1) are well defined. ...
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13 views

Differentiation of vector-function

Let $f(x) = e^{-x^Tx},$ where $x \in \mathbb{R}^n$. What will be the second derivative? The first is $~f'(x) = 2x^T e^{-x^Tx}$, and when I try to find the second, I confuse. It will be $$f''(x) = ...
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42 views

Question about vector field

If I want to draw the lines of force to a vector field F and consider whether any results are physically reasonable, then i can start with to solve the differentiable equation $$ y'(x)= {F_y(x,y) ...
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29 views

Covariant Derivative to study Holonomy

Can someone provide an explicit definition to the $\mathbb{R}^{3}$-covariant derivative? For instance, I don't understand the following calculation. Let $$E_1 = (-\cos(u) \sin(v), - \sin(u) \sin(v), ...
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1answer
27 views

Line integrals and vector fields

We had this example in class the other day, and the professor didn't not walk through how he obtained it. Compute $\int_C \vec f \cdot d\vec r = \langle 4x^3y^2 - 2xy^3, 2x^4y - 3x^2y^2 + ...
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2answers
97 views

Why is $\operatorname{Div}\big(\operatorname{Curl} F\big) = 0$? Intuition?

Why is $\operatorname{Div}\big(\operatorname{Curl} F\big) = 0$? Is there an intuitive explanation to what this means as well as an algebraic proof? Also I understand that $\operatorname{Curl} ...
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2answers
34 views

Green's Theorem and Divergence (2D)

I am reading the book Numerical Solution of Partial Differential Equations by the Finite Element Method by Claes Johnson. In Chapter 1 he talks about the Possion Equation, and to prove that FEM ...
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1answer
31 views

Curl of a function with only angular dependence

Let a function in spherical coordinates $$\vec F(\vec r) = \int{ d^3\vec r\,' \vec j(\vec r\,') } \,e^{-ik\hat r \cdot \vec r \,'}$$ Where $\vec j$ is a vector function. So $\vec F$ only depends on ...
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27 views

Stoke's Theorem for an open cylinder

How do you use Stokes' Theorem to calculate the surface integral over a cylinder of $\nabla \times F$? Do you have to calculate the line integrals along the top and the bottom? If so, is this example ...
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1answer
37 views

How to prove that the velocity field $u$ is always $\nabla\cdot u=o$ except at the origin? [closed]

$u(x)=- \frac{Q}{4 \pi} \nabla ( \frac{1}{r})$ How do you prove that the velocity field $u$ is always $\nabla\cdot u=o$ except at the origin?
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64 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 ...
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37 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 ...
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1answer
29 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 ...
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1answer
46 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 ...
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2answers
66 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?
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33 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 ...
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1answer
39 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 ...
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2answers
72 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?
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52 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 ...
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0answers
91 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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47 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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1answer
34 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 ...
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0answers
20 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
28 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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26 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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1answer
114 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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2answers
67 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
31 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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2answers
51 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
30 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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18 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 ...
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1answer
28 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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32 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
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
40 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
31 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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0answers
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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0answers
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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2answers
176 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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0answers
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
36 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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52 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
40 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
55 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 ...