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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2
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1answer
10 views

Symmetry of Green's function on the general case

Let's consider the differential equation $$\nabla\cdot[p(\mathbf{r})\nabla u(\mathbf{r})]-s(\mathbf{r})u(\mathbf{r})=-f(\mathbf{r}).$$ I want to show that the Green's function is symmetric, so that ...
-1
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0answers
24 views

Gauss and Stokes Theorem Problem!! Help!! [on hold]

In $(x, y, z)$ space is considered the vector field $V(x,y,z)=(y^2 z, yx^2, ye^y)$ solid spatial region $Ω$ is given by the parameterization: $\left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] ...
2
votes
0answers
32 views

Question about simply connected regions in $\mathbb{R}^2$

If given the vector field $\mathbf{F}(x,y)=\langle\frac{1}{x}+2xy,\frac{1}{y}+x^2-\cos{y}\rangle$, it is clear that the component functions are continuous everywhere expect at the origin. Now, when ...
1
vote
0answers
29 views

Relationship between Möbius transformations and flows/vector fields

I've noticed that the pictures illustrating the effect of Möbius transformations on the Riemann sphere (after stereographic projection to the plane) resemble the phase portrait of a vector field. For ...
1
vote
0answers
21 views

Evaluate the integral $\iint \operatorname{curl}(yi+2j)\cdot n \, d\sigma $

Evaluate the integral $\iint \operatorname{curl}(yi+2j)\cdot n \, d\sigma $ where $\sigma$ is the surface in the first octant made up of part of the plane $2x+3y+4z=12$ and triangular in the ...
0
votes
1answer
26 views

Find a closed path $C$ such that $\oint_C {\bf F} \cdot d{\bf r} \neq 0$, where $F = (y^2,x,0)$

Consider the vector field $${\bf F}=(y^2,x,0).$$ Find a closed path $C$ such that $$\oint_C {\bf F} \cdot d{\bf r} \neq 0 .$$ My attempt: I decided to try with the unit circle however the ...
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0answers
15 views

How to take partial derivative of a vector matrix vector multiplication?

I am trying to understand the mechanics of the below equations. I am especially confused about in 2.65 , how did the r.h.s which is a sum came from the gradient vector ? It would be great if someone ...
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2answers
30 views

Differentiation of $x^TAx$

I have in my text that if I differentiate $x^TAx$ with respect to the vector $x$ I get $2xA$ - could I ask why? Here $x$ is a $3\times1$ vector, $A$ is a $3\times 3$ matrix - I am given the ...
3
votes
1answer
36 views

Helmholtz decomposition of a vector field on surface

Does it make sense to do Helmholtz decomposition of a vector field defined on a surface or on a manifold? I am mostly interested in the surface case. I was trying to find a reference for this and ...
0
votes
1answer
24 views

Gradient Chain Rule: Applying Gradient in the case of a Series of Matrix operations (Neural Net Gradient Calculation)

I have the following situation: I need to calculate the gradient of the Error of a CNN a few layers deep by hand. Starting with the Error function, The $\operatorname{Error}[readoutX]= -\sum_i ...
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1answer
15 views

Show $2xx' + 2yy' + 2zz' = 0$ for curve on sphere.

This is from Manfredo P. do Carmo's Differential Geometry of Curves and Surfaces. I have just been introduced to orientations of regular surfaces, the Gauss map and the differential of the Gauss map, ...
1
vote
1answer
23 views

Find the curl of the vector field G

Find the curl of the vector field: $\underline{G}=(8r^7x-5r^3y)\underline{i}+(-8r^7y+5r^3x)\underline{j}$ where $r=(x^2+y^2)^\frac{1}{2}$ Since r is in the vector field, does it require calculation ...
1
vote
1answer
26 views

Solving constrained Euler-Lagrange equations with Lagrange Multipliers (Geodesics)

I'm trying to solve a calculus of variations geodesics problem using Lagrange Multipliers, showing that the geodesics of a sphere are the so-called great circles. I am using a constrained Lagrangian ...
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0answers
26 views

Dude with Lagrange Multipliers [closed]

Good morning, i have a problem, i don't understant very good how it work lagrange multipliers. I working in a problem, but i don't know found the $f(x,y)$ equation and the restriction. The problem: ...
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0answers
16 views

Is differentiation with respect to a vector always defined componentwise?

When one takes the derivative of a function $f$ along the direction of some vector $\mathbf{v}$, i.e. the directional derivative of $f$ along $\mathbf{v}$ this operation is defined componentwise, i.e. ...
1
vote
1answer
24 views

Finding the Gradient of a Vector Function by its Components

In Multivariable Calculus, we can easily find the gradient of a scalar function (producing a scalar field) $f : \mathbb{R^n} \to \mathbb{R}$, and the gradient function would produce a vector field. ...
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0answers
19 views

integral of partial derivative for continuity equation

Task in question is deriving velocity parallel to z component in continuity equation. $$ \frac{\partial n}{\partial t} + \left(\nabla \cdot n \textbf{u}\right) = 0 $$ $$ \frac{\partial n}{\partial t} ...
0
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1answer
24 views

Levi civita and kronecker delta properties?

I'm trying to grasp Levi-civita and Kronecker del notation to use when evaluating geophysical tensors, but I came across a few problems in the book I'm reading that have me stumped. 1) ...
0
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0answers
14 views

Line integral of $F = \langle xz, xy , 3xz\rangle $ [closed]

Let $F = \langle xz, xy , 3xz\rangle $ be a vector field. Let $c$ be the boundary of the plane $2x + y+z =2$ in the $1$st octave, counterclockwise from above. Then how can I compute the line integral ...
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0answers
21 views

Necessity of $C^{1}$ hypothesis in fundamental theorem for line integrals

The statement for the fundamental theorem for line integrals I have in my (unpublished) textbook is: Let U ⊆ Rn be an open set, let φ : [a,b] → U be a piecewise smooth curve, and let $Ω = C_{φ}$. Let ...
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0answers
13 views

i have posted a picture of the question . i can't do the second part and third part [closed]

enter image description herethe lines l1 and l2 have vector equations , I've shown that they intersect . i can't do the second and third part
0
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0answers
19 views

Surface integral of prism

I have a prism bounded by x=0, y=0, y=1-x, z=0 and z=2, and the field $v=(3x^2,xy^2,0)$ and i want to find the flow rate out of this prism. I've already figured out that only the side on y=1-x is not ...
2
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0answers
22 views

Vector Calculus: solution to Poisson equation

This is problem 8.4.17. from Marsden Vector Calculus book. Let $\rho$ be a continuous function which vanishes outside a 3D region $W$. Define ...
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0answers
6 views

How can I write a gradient of sobel filter in continuous formula?

Let $*$ denote a convolution operation, $G$ denote a kernel, and $I$ is a given image. The gradient of the image $I$ is equivalent: $\nabla (G*I) = (\nabla G) * I$ The Sobel filter approximtes two ...
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0answers
20 views

Gradient of function f(x,y,z)=h

I have function $f(x,y,z)=h$. Function $f$ is linear and I have three points $A$, $B$, $C$ given by $x,y,z$ and $h$. How can I compute gradient of $f$ ? According to further use in the article I am ...
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0answers
38 views

Volume enclosed by Implicit Surface

I am trying to calculate the Volume that's enlcosed by the surface: $$(x^2 + y^2 + z^2)^2 = xyz$$ The following is what i tried. I rewrote it in spherical coordinates where $x=r \cdot \sin\vartheta ...
0
votes
1answer
23 views

Stoke's Theorem to evaluate line integral of cylinder-plane intersection

I want to use Stokes' Theorem to evaluate the line integral $F\cdot dr$ $F = (-y^2, x, z^2)$ and $C$ is the curve of the intersection of the plane $y+z=2$ and the cylinder $x^2+y^2=1$. $C$ should be ...
1
vote
1answer
58 views

Show that ∇· (∇ x F) = 0 for any vector field [duplicate]

To solve this question, how do I define any vector field $F$, in order to solve it? I called $F = (ax,by,cz)$, in which case already $\nabla\times F = 0$. How would i go about proving this? Many ...
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votes
2answers
64 views

Parametric Surface

A surface is given by $$r(u,v) = \langle u, v^2, uv\rangle$$ (a) Evaluate the unit normal vector, $\vec n$, to the surface at the point corresponding to $u=2$ and $v=1$. I've done this by ...
1
vote
1answer
26 views

Using Stokes' Theorem to find the line integral

I am having a bit of trouble understanding line integrals. I've muddled my way through a lot of them, but I just can't understand their relation to Stokes' theorem. Here is a question that I've ...
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vote
0answers
42 views

Calculation of Variational Derivative

Following is from Olver's book on Lie groups and Differential Equations: Define the variational problem: \begin{eqnarray*} \mathcal{L}= \int_\Omega L(x,u^{(n)}) \end{eqnarray*} where $u^{(n)}$ ...
3
votes
2answers
28 views

What does it mean to use levi civita symbol with Poisson brackets in this way

I'm doing some studies in mathematical methods for physics and I came across something that I don't really understand. I have only been using the $\epsilon_{ijk}$ when I cross some vectors or ...
0
votes
1answer
11 views

Function Composition, Derivatives, Gradient, Hessian

Here's the problem: Let $f : R^n \to R$ be a twice continuously differentiable function. Let $\phi(t) = f(u + td)$ be a composition function from $R$ to $R$, with given vectors $u, d \in R^n$. ...
0
votes
1answer
46 views

Outward flux of a vector field through a rectangular box.

Let $R$ be the rectangular box consisting of all points $(x,y,z)$ with $-1\leq x\leq 1$, $-1\leq y\leq 1$, and $-1\leq z\leq 1$. Define the vector field $$V= ...
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0answers
23 views

Is the Divergence of Curl equal to Zero for All Coordinate Systems?

Is the divergence of curl equal to zero for all coordinate systems? Even a curvilinear coordinate system such as double spheroidal coordinates?
2
votes
1answer
74 views

What is the most general/abstract way to think about Tensors

In their most general and abstract definitions as Mathematical Objects : A Scalar is an element of a field used to define Vector Spaces A Vector is an element of a Vector Space. Since a Scalar ...
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0answers
26 views

Show that the Hessian of $f$ is negative definite.

Problem says: Show that if $f:A\subset \mathbb R^2 \rightarrow \mathbb R$ has a critical point $x_0 \in A$ and we let $\Delta =$determinant of Hessian of $f$ be evaluated at $x_0$, ...
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votes
0answers
21 views

Finding Directional Derivatives with gradient

Find the derivative of the function at $P_0$ in the direction of $u$.$$f(x,\, y,\,z) = \tan^{-1}\left ( \frac{5x}{9y+2z} \right ),\,\,\, P_0(7,\,0,\,0),\,\,\, u = 12i - 3j+4k$$ I understand how to ...
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0answers
15 views

Approximating the line integral

I am solving a series of problems that begins with, suppose curl $\vec{F}=\langle 5,4y,-2z\rangle$ and $C$ a circle of radius .005 centered at (2,4,5) in the plane $x+y+z=11$. The first part of the ...
0
votes
1answer
33 views

Proving ${\displaystyle{\int\!\!\int_{D}\!\!u\Delta udA<0}}$ where $D$ is the closed, unit disc.

Hello again guys and gals! I'm stuck on a problem that I thought was going to be simple for me to prove, but I was wrong (yet again). I will try not to be so long-winded this time (see my previous ...
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0answers
10 views

Vector Calculus Divergence Theorem Textbook Answer Confusion

here's a particular question I'm working on that the textbook doesn't have the same answer as me. Use The Divergence Theorem for: $F = |r|r$, where $r = <x,y,z>$, and $S$ consists of the ...
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0answers
11 views

What is this operator? (Three times curve integral)

What is this operator: https://help.libreoffice.org/File:Fo21611.png I have been seeing it in text-edit documents, but never found any explanation to it. I guess that it is a closed curve integral ...
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0answers
16 views

Gradient Application

Scalar field given as $\varphi =f\left( x,y,z\right) =x^{2}-y^{2}z$ a) Find the gradient at point (1,1,1) b) Find the partial derivative of the field at the point (1,1,1) in the direction of ...
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votes
1answer
11 views

Drawing a geometric conclusion from the curvature and torsion of a curve

If I was working with a curve $\tilde{c}(t)$ and found that the curvature $\displaystyle \kappa(t) = \frac{1}{8\sin(\frac{t}{2})}$ and the torsion $\tau(t) = 0$. What geometric conclusion, about the ...
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0answers
39 views

Working with a vector field

Consider the vector field $\tilde{F}(x,y,z) = ax \tilde{i} + by \tilde{j} + cz\tilde{k}$ (1) Let $\tilde{c}(t) = (x(t), y(t), z(t))$ be the flow line such that $\tilde{c}(0)= (x_0, y_0, z_0)$. Find ...
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0answers
61 views

Computing the vector Laplacian in spherical coordinates using metric tensor

I want to compute the Laplacian of a vector in spherical coordinates using metric tensor. I started only with the first component but I have obtained a different formula! Let be $\vec{v} $ a vector ...
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0answers
29 views

Vector field, flow line question. Need help please

Consider the vector field $\tilde{F}(x,y,z) = ax \tilde{i} + by \tilde{j} + cz\tilde{k}$ Let $\tilde{c}(t) = (x(t), y(t), z(t))$ be the flow line such that $\tilde{c}(0)= (x_0, y_0, z_0)$. Find ...
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0answers
26 views

Capacitance and Gauss' Law

If the area of a single plate is $A$, show that the capacitance $C$ = $\frac{q}{v}$ is directly proportional to $A$ but inversely proportional to $d$. You may use Gauss' Law: $\nabla$$\cdot$$E$ = ...
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0answers
17 views

Is it d^3V or dV when integrating for volume (triple integral)?

In a volume integral dxdydz is shortened to dV. I have also seen d^3V. So dcubed V. What is the correct way to write it, or are both correct?
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1answer
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Torque when system is constrained to rotate about $\vec{r}$ [closed]

Let $\vec{F}, \vec{r}$ and $W$ are elements of $R^{3}$. Given $\vec{F}= -\nabla{W}$. Let system be constrained to rotate about $\vec{r}$. How can we find Torque?