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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3
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2answers
31 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
15 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
54 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= ...
0
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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?
3
votes
1answer
79 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 ...
0
votes
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 $x_0$, ...
0
votes
0answers
22 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 ...
1
vote
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 ...
0
votes
0answers
11 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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votes
0answers
12 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 ...
0
votes
0answers
17 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 ...
0
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 ...
0
votes
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 ...
0
votes
0answers
65 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 ...
0
votes
0answers
31 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 ...
0
votes
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$ = ...
0
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0answers
18 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?
-1
votes
1answer
19 views

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?
0
votes
1answer
17 views

Finding the radii of an ellipse from the intersection of a plane and a sphere

I'm trying to solve the following problem, regarding Stokes Theorem: $F = z i + xj + yk $; C the curve of intersection of the plane $x + y + z = 0$ and the sphere $x^2 + y^2 + z^2 = 1$ [Hint: ...
2
votes
0answers
20 views

Poincaré-Hopf Index Theorem - Intuitive explanation

I heard about an interesting theorem. Poincaré-Hopf Index Theorem : If $\vec{v}$ is a smooth vector filed on the compact, oriented manifold $X$ with only finitely many zeros, then the global ...
1
vote
2answers
102 views

Finding extrema points with lagrange multipliers

Using lagrange multipliers, find all the extrema points of the function $f(x,y) = x^2 + (y-b)^2$ subject to the constraint $y = x^2$. Using the fact that critical points occur at $\triangledown ...
1
vote
1answer
67 views

Kinetic energy of incompressiblue fluid

I am trying to show that the kinetic energy for an incompressible and irrotational fluid with no sources and no sinks is given by $$\frac{\delta}{2} \iint_{S} \psi \frac{\partial \psi}{\partial n} ...
3
votes
1answer
33 views

Confused on surface integral problem

I am asked to evaluate $$\iint_{S} [ \nabla \phi \times \nabla \psi] \bullet n dS$$ where $\phi=(x+y+z)^2$ and $\psi=x^2-y^2+z^2$ where S is the curved surface of the hemisphere $x^2+y^2+z^2=1$ , $z ...
1
vote
1answer
23 views

question on vector calculus notation

I just have a question about the vector calculus notation: $$(u \cdot \nabla)u$$ Is that the same as $( \nabla \cdot u)u$?
0
votes
0answers
15 views

Circulation around a curve

in a previous question I had asked How to apply the divergence thereom in the plane About the divergence of $F=(xy)i+(2x-y)j$ where C is the triangle with verticies $(0,0)$ , $(1,0)$ and $(0,1)$. ...
0
votes
1answer
34 views

Gradient of curvature over triangle

In an article "Smooth Feature Lines on Surface Meshes", there is this in Equation (4): ∇Ki(T), i = {min, max} where K is principal curvature and T is a triangle. How do I calculate this?
0
votes
1answer
38 views

vector calculus using Lagrange Multipliers

$(1)$ Let $c\in R$ be a constant. Using Lagrange Multipliers, find all the extrema of $$f(x,y) = x^2 + (y-c)^2$$ subject to the constraint $$y = x^2$$ I'm pretty sure I've found the critical ...
0
votes
1answer
11 views

Vector field line integral: confusion about sign of dl, order of limits

I have some confusion about simple line integrals of vector fields. If I want to calculate integral $\int_C\underline F\cdot d \underline l$ from point $(1,0)$ to $(0,0)$ in a straight line, then ...
0
votes
1answer
16 views

How to show this vector cross product/gradient result

One of my books has that if $$\bar A= \phi \nabla \psi$$ then $$\nabla \times \bar A = \nabla \phi \times \nabla \psi$$ But I don't see why it is true. What is the proof of this? Thanks
1
vote
1answer
31 views

Line integral using Green's Theorem considering equation of ellipse

I am given the integral $$I=\int_{C} \frac{y}{4x^2+7y^2} dx - \frac{x}{4x^2+7y^2}dy$$ where C is the rectangle with vertices $$A=(4, 7), B=(-4, 7), C=(-4, -7), D=(4, -7)$$ oriented in the ...
0
votes
1answer
17 views

Vector difference norm bound of $\|\frac{x_1}{\|x_1\|^2} - \frac{x_2}{\|x_2\|^2}\|$

Given $\|x_1 - x_2 \| \leq C$ where C is a constant, could we derive a bound of $\|\frac{x_1}{\|x_1\|^2} - \frac{x_2}{\|x_2\|^2}\|$?
1
vote
1answer
35 views

Evaluating Line Integral with Green's Theorem

I'm given a line integral $$\int_{C} \left(\frac{\sin(3x)}{x^2+1}-6x^2y\right) dx + \left(6xy^2+\arctan\left(\frac{y}{7}\right)\right) dy$$ where C is the circle $$x^2+y^2=8$$ oriented in the counter ...
0
votes
1answer
27 views

lim r(t) = L implies lim ||r(t)|| = ||L||

The question is: if $\lim_{t\rightarrow t_0}\vec r(t) = \vec L$ show that $\lim_{t\rightarrow t_0} \|\vec r(t)\| = \|\vec L\|$ So here's where I am so far: let $\vec r(t) = (f_1, f_2,...f_n)$ be ...
1
vote
3answers
31 views

Work Done when more than one field exist

Suppose we have two different electric field, $\vec{E_1}$ and $\vec{E_2}$ where $\vec{E_i}$ are elements of $\mathbb R^2$ $y>0 => \vec{E}$=$\vec{E_1} $ and $y<0 => \vec{E}$=$\vec{E_2} $ ...
2
votes
0answers
31 views

Green's theorem with unit normal and del operator

By appropriately choosing the functions P and Q in Green's theorem, show that $\iint_R\nabla^2 \phi\;dA =\int {\partial \phi}{\partial n} \;ds $, where $\frac{\partial}{\partial n}$ denotes ...
0
votes
1answer
110 views

How to evaluate circulation

I am having some trouble with the following question: Solve for the circulation in five different ways for the velocity field $$\bar v=(e^{-x^2}-yz)\hat i+(e^{-y^2}-xz+2x)\hat j+(e^{-z^2})\hat k$$ ...
1
vote
1answer
33 views

Differentiation that involves determinant and vectors

How can i calculate following? $\frac{d}{dR'} {(|\vec{R}-\vec{R'}|)^{-1}}$ I tried this but I'm not sure this a valid method: let $\vec{R}-\vec{R'} = \vec{u}$, then $-\vec{dR'}= \vec{du}$ ...
1
vote
0answers
18 views

How could I determine the form of a function that chases another function?

This is a problem that a teacher told me about that's been bothering me for a while. I'm positive that this has been explored before because it seems way too useful for physicists to not have come up ...
1
vote
1answer
17 views

Neumann condition for Poisson equation

Solving $ \nabla^2u = 1 $ for spherically symmetric u in the region $r < a, a > 0$, with the following conditions at r = a (separately) (a) $u = 0$ (b) $\nabla u \cdot n = 0 $ where n is the ...
0
votes
1answer
42 views

How to apply the divergence thereom in the plane

I am trying to work a seemingly simple practice problem but I am having some confusion. The question asked to verify the divergence theorem in the plane for the vector field $$F=(xy)i+(2x-y)j,$$ where ...
0
votes
0answers
16 views

Global Clebsch potentials

For an aribitrary vector field $\mathbf{v}$ on $\mathbb{R}^3$, it always can locally be written as $$ \mathbf{v}=\nabla f+g\nabla{h} $$ where $f$, $g$, $h$ are called Clebsch potentials. My question ...
2
votes
1answer
26 views

Trouble plotting Maple space curve given a parametrization

I am being asked to plot a curve C with parametrization $$ r(t)=\left \langle \sin(mt)\cos(nt), \sin(mt)\sin(nt), \cos(mt) \right \rangle $$ with parameters of $$0\leq t\leq 2\pi$$ with integers ...
1
vote
1answer
26 views

Find the volume of ice cream cone using cylindrical/spherical coordinates

I'm stuck on what the boundaries are for the volume bounded by the cone $z=-\sqrt{(x^2+y^2)}$ and the surface $z=-\sqrt{(9-x^2-y^2)}$ $\,\,$-essentially an upside down ice cream cone Remember that ...
1
vote
0answers
33 views

Confusion with vector addition

So while reading a solid-state physics book, I encountered the following (drift-diffusion equation): I know the divergence of a vector field produces a scalar field, and so the second equation is ...
0
votes
0answers
33 views

Solving for $G(x,y)$ in a Gradient System (Differential Equations)

If I'm given the following, how would I solve for $G(x,y)$? $$\begin{align}\frac{dx}{dt}&=y^2-\cos x\\ \frac{dy}{dt}&=2xy-\sin y\end{align}$$ I know $x'$ is equal to the partial of $G$ with ...
0
votes
1answer
26 views

Change order of integration

I'm stuck on how to change the order of integration for this question, $$\int_0^{2\pi} \int_{\cos x}^1\, f(x,y)\,\,\,dydx $$ We know that if we take vertical strips of the original integral the ...
0
votes
0answers
19 views

Find constants so that the directional derivative of $f(x,y,z) = axy^2+byz+cx^3z^2$ has maximum value $32$ in point $P$ given the direction

I am asked to find $a$, $b$ and $c$ so that the directional derivative of $$f(x,y,z) = axy^2+byz+cx^3z^2$$ has maximum value of $32$ in the point $P(1,2,-1)$ and in the direction $\overrightarrow{u} = ...
0
votes
2answers
42 views

Express the vector field in polar coordinates

$\underline{G}=(8r^7x-5r^3y)\underline{i}+(-8r^7y+5r^3x)\underline{j}$ where $r=(x^2+y^2)^\frac{1}{2}$ How would I express the vector field as cylindrical coordinates? I have looked at various ...
0
votes
1answer
34 views

How can I solve this problem? [closed]

A rotation φ1 + φ2 about the z-axis is carried out as two successive rotations φ1 and φ2, each about the z-axis. Use the matrix representation of the rotations to derive the trigonometric identities