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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34 views

Biot-Savart Law to construct vector potential for divergence free vector field on $\mathbb{R}^3$

I would like to confirm a method I am trying to use which uses the Biot-Savart Law to construct a vector potential $\underline{w}$ for a divergence free vector field $\underline{v}$ on $\mathbb{R}^3$. ...
1
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1answer
35 views

Euclidean norm on integer lattice

Does the Euclidean $ L^2 $ norm (and distance) make any sense on an integer lattice in $ \mathbb{R}^n $? And what is the preferable way of calculating a type of norm in such spaces?
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1answer
55 views

Gradient of Function

I am trying to find the $\nabla F$ with respect to $x_i$ where $F$ is as follows: $$F(x_0,...,x_n) = c_1\sum_{i=0}^{n}\sum_{j=1}^{k}\frac{1}{||x_i - r_j||_2^2} + c_2\sum_{i=0}^{n-1}||x_{i+1}-x_i||_2^...
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0answers
22 views

A continuous, path-independent, non-conservative vector field

I am trying to define a non-conservative vector field on $\mathbb{R}^2$ whose line integrals don't depend on its paths. I think it could be $$ F(x,y) = \begin{cases} (x+y, x-y) && \mbox{if $(x,...
4
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0answers
91 views

Uniqueness of a PDE solution

Suppose $F:U \to \mathbb{R}^n$ is a $\mathcal{C}^1$ vector field on an open set $U \subseteq \mathbb{R}^n$. Let $\lambda \in \mathbb{C}$. Consider the partial differential equation $$F(x) \cdot \...
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0answers
36 views

Verification of divergence theorem

At time t the velocity field of a fluid is given by $\bar V(x,y,z)=x^3 {\bf i}+y^3{\bf j}+z{\bf k}$ the outward flux integral $ Φ = \iint_S \bar{V}\cdot d\bar{S} $ where S is the surface of the ...
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0answers
23 views

solving linear gradient PDE

Let $f:\mathbb R^n \to \mathbb R$ be a function that satisfies the equation: $$\nabla^T f = - (\nabla^T \phi ) f $$ where $\phi:\mathbb R^n \to \mathbb R$ is some given function and $\nabla^T$ is the ...
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1answer
55 views

An Example Where $Df$ Is Not Integrable

I have been trying to find an example where $f$ is continuous on $[0,1]$ and is differentiable on $(0,1)$, however, $Df$ (gradient) is not integrable on $[0,1].$ I thought of one example: $f(x,y)=...
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0answers
18 views

Modeling the Motion of a Particle where $ ||\vec{f_i}|| = \frac{k_i}{r_i^2} $

At the origin of an $n$-dimensional space, there exists a single free-moving particle ($\gamma$) with a known mass ($m$) and velocity ($\vec{v}$). There also exists $p$ number of fixed points with ...
1
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1answer
48 views

Evaluate Path Integral

Consider the vector field F=\begin{pmatrix} -3z^2sinx \\[0.3em] 8y^3z \\[0.3em] 2y^4+6zcosx \end{pmatrix} By evaluating the path integral, compute $\...
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2answers
24 views

Finding the points on the curve where the gradient of the tangent is equal to 2.

So, my question is as following. "Find all the points on the curve $y= 2x^3 + 3x^2 - 10x +3$ where the gradient of the tangent is 2." I used $\frac{dy}{dx}$ in order to get the formula for the ...
3
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1answer
74 views

How do we calculate the gradient from numerical data

I have a 3D irregular mesh. Each coordinate point holds a specific data, say , for example, temperature. For a specific point, how do I calculate the temperature gradient ?
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2answers
22 views

Continuity and differentiability of a function defined parametrically

How do we check continuity and differentiability of a function defined parametrically e.g. $$x=2t-|t-1|$$ and $$y=2t^2+t|t|$$
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2answers
34 views

Gradients of two functions with the same level sets are parallel for all points in the intersection of the level sets

Is the intuition behind this statement, based on : 1. The definition that a gradient is perpendicular to the level curves 2. Since the level sets are the same for both functions, the corresponding ...
3
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1answer
60 views

Prove that $f( E^o)$ is an open set if $E$ is bounded

I do not really understand how to proceed with this question; To prove a), do I need to show that every point in $f(E^0)$ is an interior point? I would greatly appreciate any help in this regard
2
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1answer
48 views

The time derivative of the absolute value of a gradient.

I am interested in finding out the time rate of change of the absolute value of the density gradient, such that the directional change of the density gradient does not affect the final sign of the ...
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0answers
32 views

Is there a nuclear norm approximation for stochastic gradient descent optimization?

I want to minimize $E$ by using stochastic gradient descent. I know that there is a sub-differential for the nuclear norm, but i want to know if is there a approximation of nuclear norm in order to ...
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0answers
20 views

What is the appropriate way to average a set of slopes?

I have a set of line slopes (output from linear regressions) and I'd like to combine them to yield some measure of the 'slope' of the set. However, simply taking the mean of the numbers doesn't make ...
0
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1answer
31 views

Applying the mean value theorem for multivariate functions

In the Salas Calculus book (page 805 in the 10th edition) they say that $g(t) = f(\mathbf{a} + t[\mathbf{b} - \mathbf{a}]), t \in [0,1]$. I get that the point is to find an equivalent single variable ...
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0answers
25 views

Maximum normal derivative of a function at a point

Question: Calculate the maximum normal derivative of the function $$\phi(x,y,z)=yz+zx+xy$$ at $P(1, 1, -2)$. I have solved problems on maximum directional derivative. But what is maximum normal ...
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0answers
26 views

The difference between steepest descent and gradient decent

It is known that gradient descent and steepest descent method are the same. but I found(on Convex Optimization by Stephen Boyd Page 490) that there is difference between them? The direction of ...
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0answers
64 views

Prove topological space has countable basis

Given a topological subspace M of $\mathbb{R}^2 \times S^1$ defined by $(x,y,e^{i\theta})$ and two charts $(U,h)$, $(V,k)$ such that $H:\mathbb{R} \times (-\pi,\pi) \to M$ $H(x,\theta) = (x,x\tan(\...
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2answers
40 views

If $|\vec a|=12$ and $|\vec b|=4\sqrt 3$ and $\vec b.\vec c=24$ , then find $|(\vec a \times \vec b) +(\vec c \times \vec a)|$

Let $\Delta PQR$ be a triangle. Let $\vec a = \vec {QR}, \vec b = \vec {RP}$ and $\vec c = \vec {PQ}$. If $|\vec a|=12$ and $|\vec b|=4\sqrt 3$ and $\vec b.\vec c=24$ , then find $|(\vec a \times \vec ...
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0answers
10 views

Showing matrix norm inequality involving auxiliary functions holds in Higher Dimensional Chain Rule Proof.

I wanted to prove the Chain Rule in a higher dimensional setting: Let there be two mappings: $ \varphi: \mathcal{U} \to \mathbb{R}^{n} $ where $ \mathcal{U} \in \mathbb{R}^{m} $ is open $ \psi: \...
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1answer
16 views

Formulation of a vector field

Let $V=y\,\partial/\partial x+x\,\partial/\partial y$ be a vector field on $\mathbb{R}^2$. I don't understand about how the vector field $V$ is defined. For example how do we map a specific point (...
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0answers
45 views

Helmholtz-Decomposition-weak Formulation in n-dimensional case

In the Wikipedia article about the Helmholtz-Decomposition it says in the section about Weak Formulation: For a slightly smoother vector field u ∈ H(curl, Ω), a similar decomposition holds: $\...
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4answers
38 views

Functions of multivariables

I'm having issues with calculating: $\lim (x,y)$ approaches $(0,0)$ for $$f(x,y)=\frac{x^2y}{2x^2 +3y^2}$$ Can someone help me please?
2
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0answers
30 views

Evaluate $\int \int \vec{f} . \hat{n} \ \ ds$ using stokes theorem

Consider $\vec{f} = (2x-y)\hat{i} - yz^2 \hat{j} -yz^2 \hat{k}$ where $S$ is the upper half of the surface of the sphere $x^2 +y^2 +z^2 = 1$ and C is its boundary $Stoke's$ Theorem :Let $S$ be a ...
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0answers
13 views

Vector Triple Product (Coefficient I'm missing) Question [duplicate]

Simple question. Why is it this: $\mathbf{A} \times (\mathbf{\nabla} \times \mathbf{A}) = \mathbf{\nabla}(\frac{1}{2}|\mathbf{A}|^{2}) - (\mathbf{A} \cdot \mathbf{\nabla})\mathbf{A}$ and not this: ...
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0answers
53 views

irrotational vector-field => Existence of scalar potential - for Sobolevfunctions

For $v\in C^1(\Omega,\mathbb{R}^n)$ the following is well-known: Let $\Omega \subset \mathbb{R}^n$ be simply connected. Then for every $v\in C^1(\Omega,\mathbb{R}^n)$ with $curl \;v = 0$ there exists ...
2
votes
1answer
91 views

Proof of $-\nabla\times\omega = \nabla^2 U$

What is a proof for $$ -\nabla\times\omega = \nabla^2 U $$ in the scope of fluid mechanics? I'm learning vector calculus for my project and stuck on this seemingly simple proof problem. Detailed ...
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0answers
32 views

divide both sides of an equation by Levi Civita symbol

$\epsilon_{ijk} \frac{\partial E_{k}}{\partial x_{j}} = - \frac{\partial B_{i}}{\partial t} $ The above is one of the Maxwell's equations - I want to get $\frac{\partial E_{k}}{\partial x_{j}} $ as ...
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0answers
34 views

$\int_{-\pi/2}^{\pi/2} |\vec{v}-\vec{V}|(\vec{v}-\vec{V}) d t$, with $\vec{v}=(\vec{i} \cos t +\vec{j} \sin t) (\sqrt{1-p^2 \cos^2 t}+p \sin t)$

I need to solve this integral: $$\int_{-\pi/2}^{\pi/2} |\vec{v}-\vec{V}|(\vec{v}-\vec{V}) d t$$ $$\vec{v}=(\vec{i} \cos t +\vec{j} \sin t) \sqrt{1-p^2 \cos 2 t+2 p \sin t \sqrt{1-p^2 \cos^2 t}}=$$ $...
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1answer
38 views

How to calculate $\vec\nabla\times\frac{\vec{r}}{r^3}$ when $r$ can be $0$?

I already know that $\vec\nabla\cdot\frac{\vec r}{r^3} = 4\pi\delta(\vec r)$. In that case, we calculate it directly when $r\neq 0$ and use Gauss' Law to prove it is equivalent to $4\pi\delta(\vec r)$....
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1answer
30 views

Show curve is tangent to a surface using gradient

The question is this: 'Show that the sphere $h(x,y,z) = x^2+y^2+z^2-8x-8y-6z+24=0$ is tangent to $f(x,y,z)=x^2+3y^2+2z^2=9$ at the point (2,1,1).' My approach was that grad(f) at P should give a ...
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2answers
96 views

Disadvantage of Newton method in optimization compared with gradient descent

In optimization with Newton method in wikipedia, there is a diagram showing Newton's method in optimization is much faster than gradient descent. What is the disadvantage of Newton's method compared ...
6
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1answer
56 views

Lie bracket of exact differential one-forms

Let $(M,g)$ be a Riemannian manifold. The musical isomorphisms $^\flat:\chi(M) \to \Omega^1(M)$ and $^\sharp:\Omega^1(M) \to \chi(M)$ allow the space of differential one-forms $\Omega^1(M)$ to be ...
0
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0answers
28 views

Gradient Descent Algorithm

Hello I'm trying to understand how the Gradient Descent Algorithm works. There is a formula that I found on wikipedia and that I cannot justify: https://www.wikiwand.com/en/Gradient_descent#/...
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15 views

Is there a nice expression for an object of the form $\nabla(\textbf{A}\cdot\textbf{B})$ in even dimensions?

I know for instance that $\nabla(\textbf{A}\cdot\textbf{B})$ can be expressed as $$\nabla(\textbf{A}\cdot\textbf{B}) = (\textbf{B}\cdot\nabla)\textbf{A} + (\textbf{A}\cdot\nabla)\textbf{B} + \textbf{...
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0answers
16 views

Vector Magnitude Equality 2

I want to show that: $$(n \cdot \nabla n)^2=(n \times \nabla \times n)^2$$ $n$ is a unit vector and $-n$ is essentially $n$ for this application. Using the vector triple product I can get the inner ...
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1answer
32 views

Vector Magnitude Equality

I'm trying to show that: $$(\nabla \times n)^2=(n \cdot \nabla \times n)^2+(n \times \nabla \times n)^2$$ I know that $n$ is a unit vector and that $-n$ is effectively the same as $n$ for the ...
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2answers
282 views

Can a gradient vector field with no equilibria point in every direction?

Suppose that $V:\mathbb{R}^n \to \mathbb{R}$ is a smooth function such that $\nabla V : \mathbb{R}^n \to \mathbb{R}^n$ has no equilibria (i.e. $\forall x \in \mathbb{R}^n : \nabla V (x) \not = 0$). ...
2
votes
1answer
45 views

Constructing a partition and sigma algebras

Let $Ω$ be a set and let $E$ $=$ {$A$, $B$} where $A$, $B$ ∈ $Ω$ and none of $A$, $B$, $A$ $∩$ $B$, $A$ \ $B$ are empty. By constructing a partition of $Ω$ based on ${A, B}$ prove that $σ(E)$has 16 ...
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0answers
13 views

An equation to figure out where on the other side of earth you are looking

I'm trying to derive out an equation that will take variables for the longitude and latitude of your current position on earth, and the pitch and heading that you are looking downwards towards the ...
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1answer
22 views

Derivative defined over nxn matrics

Let $V$ be the vector space $M_{n,n}(\bf{R})$, and suppose $g : V \to V$ is defined by $g(\mathbf{a}) = \mathbf{a}^{3}$. I'm asked to compute the derivatives of $g$ in the form $g'(\mathbf{a})(\mathbf{...
0
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1answer
28 views

coupled vector equations

So I am trying to solve some problems from Hughston and Tod's Introduction to General Relativity. I need help with the following system of equations: $$aX_i+bY_i=P_i$$ $$\epsilon_{ijk} X_jY_k=Q_i$$ ...
2
votes
3answers
41 views

Vector Equation involving cross product

I am trying to solve a problem from Vector Analysis, which should be fairly easy, but somehow I can't solve it. Solve for $X_i$ $$kX_i+\epsilon_{ijk}X_jP_k=Q_i$$ Also I am trying to solve the ...
2
votes
2answers
54 views

Curl of a Point Vortex Flow and its Circulation

I have the following 2D vector field $U=(u,v)=\frac{1}{x^2+y^2}(y,-x)$. When taking the curl of this field it returns zero. But when I take the circulation of the field defined as $$\Gamma=\oint_C U\...
2
votes
1answer
53 views

Divergence theorem in curvilinear coordinates

Suppose I have a tensor \begin{gather} \stackrel{\leftrightarrow}{A} = \begin{bmatrix} a_{11}(\vec{r}) & a_{12}(\vec{r}) & a_{13}(\vec{r})\\ a_{21}(\vec{r}) & a_{22}(\vec{r}) & a_{23}(...
1
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0answers
20 views

Integral of divergence over a closed surface

I am reading a paper, where an integral of a divergence over a closed surface is used without proof. $\oint_S \nabla \cdot \vec{v}(\vec{r}) = 0$, where $\vec{v}$ is tangential to the surface ($\vec{...