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Questions tagged [vector-analysis]

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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How to transfer a factor in vector field to the density when facing with divergence?

This question comes from the following passage, Since the divergence is defined in terms of volume form, I wonder how it is with the density. Also, I dont understand how the factor is transferred... (...
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2answers
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Continuity for a two-variable function from baby Rudin

While I'm studying Baby Rudin's Exercise, I got some problems. The Exercise $9.28$ says I need to check the continuity of the function $$\varphi(x,t)=\begin{cases}x&0\leq x\leq \sqrt{t} \\ -x+2\...
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1answer
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Angle between two intersecting lines that appear on a cut face

I am given the equations of 2 thin planes in a rock. Plane 1: $\mathbf r\cdot(0,3,-1) = -1$ Plane 2: $\mathbf r\cdot(1,0,1) = 1$ The entire set (the rock) is cut by the plane: $\mathbf r\cdot(-1,3,...
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1answer
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How to graph $\eta(t)=(t,|t|)$?

We are studying lines (I am not quite sure about the correct term in English and could not find any other term for this from googling so please correct me if I am wrong) in our calculus class right ...
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2answers
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How to calculate the gradient (vector) of a vector field?

First question from me. I hope it's clear enough. I'm trying to make a physics simulator of magnets, magnetic interactions and such. I've followed the formulas in Wikipedia pages, and some first ...
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A Stokes' Theorem problem.

I know this is Stokes' Theorem. How should I proceed to proof that by dotting a constant vector on both sides I get the answer . $\phi$ is a scalar. Prove $$ \oint_C\phi\,\mathrm d\mathbf r = \...
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1answer
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Gradient direction

I learned at the math classes that the gradient at a point is perpendicular to the surface but our electromagnetism teacher taught us that the gradient is tangent to the graph and it points in the ...
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1answer
28 views

Time Derivative of Kinetic Energy in Fluid Dynamics

I'm trying to work out the value of $dT/dt$ where $T$ is the total kinetic energy of the flow and $t$ is time, for a fluid motion with velocity $\underline{u}$, pressure $p$ and constant density $\rho$...
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1answer
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Usage of Lagrange Multipliers in multivariable calculus

I just learnt about Lagrange Multipliers & am confused about why they are useful. Why can we not just check for critical points by checking if the gradient vector of the objective function $f$ is $...
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1answer
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Differentiating an arbitrary vector function with angular momentum operator in quantum mechanics

This is part of a quantum mechanics problem, where the eigenvalues of angular momentum operator in 3D are supposed to be calculated. The 2 particle wave function is given as: $$\Psi(\textbf{r}_\...
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Question about Bidirectional Associative Memory(BAM) testing.

I have designed a BAM for the recognition of the 4x5 binary images. (total of three images) I was testing it, first i gave it inputs ( Xi, where i=1,2,3 etc.) and it gave correct results (meaning it ...
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How to compute derivative/gradient of matrices? E.g. $w^T X^T Xw - 2w^T X^T t +t^Tt$?

How to compute derivative/gradient of matrices? E.g. $w^T X^T Xw - 2w^T X^T t +t^Tt$? Intuitively $2w^TX^Tt$ looks like $2xy$ so the derivative would be $2X^Tt$. But what about $w^TX^TXw$? $X$ is $\...
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1answer
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Vector spaces - External Direct Sums

Let $U$ and $W$ be vector spaces. Define a new vector space, consisting of the set $U \times W =$ $\{(\vec{u},\vec{w})| \vec{u} \in U, \vec{w} \in W\}$ along with these operations: $(\vec{u_1},\vec{...
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1answer
33 views

Taking divergence of the gradient of a scalar field that depends only on the position vector in $\mathbb{R}^3$.

The scalar field $f$ depends only on $r=|\underline{\mathbf{r}}|$ which is the position vector in $\mathbf{R}^3$ and I need to calculate the quantity, $$\nabla \cdot \nabla f$$ i went about ...
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Potential for Monotone Operator

I have a question about understanding the proof of Theorem 4.11 in the paper A Potential Theory for Monotone Multivalued Operators (accessible here). The authors claim to construct a convex ...
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1answer
30 views

Laplacian of cross product of two vectors

Suppose A and B are two vectors then what is $\nabla^2 (\mathbf{A} \times \mathbf{B}$) I tried finding it on the internet but no luck.
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2answers
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Linear dependence after a linear transformation

Find an example for each of the following, or explain why no example can be found: a) a linear map $\phi_3 : M_{2\times 2}(\mathbf{R}) \to \mathbf{R}_4$ and matrices $\ell_1, \ldots, \ell_4 \in ...
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2answers
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derivative of a gradient

just a very simple question: I have a continuous and differentiable function $\phi$ defined on $R^3$ with its gradient $\nabla\phi$. Could you please tell me if it makes sense to take a derivative ...
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1answer
24 views

Is minimizing the norm of a vector equivalent to minimizing the square of the norm?

Are the following two problems equivalent? $$ \arg \min_{y} || A \vec{x} - \vec{y} || $$ and $$ \arg \min_{y} || A \vec{x} - \vec{y} ||^2 $$ I feel like they are equivalent as it is just the ...
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integrate a differential form over the indicated smooth cube

I'm supposed to integrate the differential form over the smooth cube and I'm not sure if I'm on the right track. So suppose, $ \int_c x dy+ y dz$ where $c: [-1,1] \rightarrow \mathbb{R^{3}}$ is $c(...
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Limit sets of a gradient field

I am trying to solve this question on J. Sotomayor's book on ODEs. Define $X=\nabla f$, $f$ being defined in an open subset $\Delta \subset \mathbb R^n$. Prove that $X$ has no periodic orbits. And, ...
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1answer
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How to calculate $\frac{\partial\Theta}{\partial L}$ if I know $\frac{\partial L}{\partial\Theta}$?

How to calculate $\frac{\partial\Theta}{\partial L}$ if I know $\frac{\partial L}{\partial\Theta}$? Suppose I have a halved sum of squared errors loss: $$L(\Theta)=\frac{1}{2}\sum^{M}(y-h(X\circ\...
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Operation on a vector with a special property

I have a vector $\vec u$ with the following property: $$\vec u (tx, ty, tz) = t^n \vec u (tx, ty, tz)$$ Now I have to prove that: $$(\vec r.\vec \nabla)\vec u = n.\vec u$$ How can I do this? I tried ...
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1answer
26 views

Derivation of the trace with Hessian matrix

Take the matrix $\Sigma\in\mathbb{R}^{n\times n}$ and the function $f:\mathbb{R^n}\rightarrow\mathbb{R}$ in $C^2$. 1) How can I compute the matrix derivation $$\frac{\partial (tr\left(\Sigma\Sigma^...
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Laplace to gradient chain rule

how do I do the chain rule for f(-$\Delta$)gh if f,g,h are functions and I want it written in terms of gradients. I know f(-$\Delta$)g= $\nabla$f x $\nabla$g.
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Solve optimization problem using KKT conditions

I'm trying to understand the solution to Boyd and Vandenberghe Problem 5.30: Boyd and Vandenberghe Problem 5.30 The Lagrangian is $$L(X,\nu)=\text{tr}X-\log\det X+\nu'\left(Xs-y\right),$$ so the ...
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2answers
39 views

Gradient of matrix $b^{T}x$

I'm trying to understand my exam solution from the lecturer, but I got confused over one small thing in the solution. Problem: Consider $f(x)=\frac{1}{2}x^{T}Ax - b^{T}x$ Where $A=\begin{bmatrix}2&...
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Exponential integral of translation invariant function

Consider function $f: (\mathbb{R}^{d})^{n} \rightarrow \mathbb{R}$ with spatial invariance property of the form : $f(x_1,x_2,...,x_n) = f(x_1 + \zeta, x_2 + \zeta,..., x_n + \zeta)$ for $\zeta \in \...
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1answer
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Norm, gradient, vector

Can I say that the norm of a component of the vector is smaller than the norm of the whole vector? (in this case the gradient) $$\|v_x\| < \|\operatorname{grad} v\|$$
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System of ODE's - Why does this vector function connect two constant vectors continuously?

Given the system of two ODE's $\frac{{d{\mathbf{w}}}}{{d\xi }} = {\mathbf{r}}({\mathbf{w}}(\xi ))$, with $\lambda ({\mathbf{w}}(\xi )) = \xi $ and ${\mathbf{w}}(\lambda ({{\mathbf{u}}_L})) = {{\...
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1answer
81 views

Convergence of Vector Combinations

I have $n$ vectors $v_1(t), \dots, v_n(t)$. Time is divided into discrete rounds. Initially, all vectors have length $\leq 1$. The vectors at the next time step $t+1$ can be calculated as follows: \...
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2answers
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How to straighten up this spiraling vector field?

Let $X(x_1, x_2) = (x_1 -x_2)\partial_1 + (x_1 + x_2 + 1)\partial_2$ be a vector field on $\mathbb{R}^2$. I want to straighten it up around point $0$ . $X(0) = \delta_2|_0 \neq 0$. So it must be ...
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Differentiating a simple, single-variable equation involving a vector

Please forgive how simple this is, but I can't seem to find any explanations for how to differentiate single-variable equations of the following form: $f(\boldsymbol{x}) = 5\boldsymbol{x}$, where $\...
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Vector operations.

At 12:00 pm, a spaceship is at position [3,2,4]km away from the origin with respect to some 3 dimensional co ordinate system. The ship is travelling with velocity [-1,2,-3]km/h What is the location of ...
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Divergent of a dot product of tensor and vector

Hei, I am trying to derive energy equation from Navier-Stokes equation and I come across this: $$\nabla.(\sigma.v)=(\nabla.\sigma).v +\sigma:\nabla v$$ $\sigma $ is the stress tensor V :is the ...
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Difference between $\int_c F \cdot dr$ and $\int_c F\times dr$?

$\boxed{Q . 1}$ If $\phi=2xyz^2$, $\vec F=xy \hat i-z \hat j +x^2 \hat k$ and C is the curve $x=t^2,y=2t,z=t^3$ from $t=0$ to $t=1$ then evaluate $\int_c F\cdot dr$ and $\int_c F\times dr$ I don't ...
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Gradient of $||Ax - y||^2$ with respect to $A$

How do I proceed to find $\nabla_A||Ax - y||^2$ where $A \in \mathbb{R}^{n\times n}$ and $x,y \in \mathbb{R}^n$ and the norm is the Euclidean norm. Attempt so far $$||Ax - y||^2 = (Ax-y)^T(Ax-y) = ...
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How Do You Find an Equation of the Tangent Plane for a Torus?

I've parameterized the torus given by $z^2 + (r - 2)^2$ = 1 using: x = (2 + $\cos\theta$)$\cos\alpha$, y = (2 + $\cos\theta$)$\sin\alpha$, z = $\sin\theta$. I'm really stuck on how to find the ...
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1answer
12 views

Line integral of a vector field along a curve C with two segments

Vector field $ \vec F = (3x^2y^3+8x)\vec i + 3x^3y^2\vec j$, along a curve C consisting of two segments C$_1$ and C$_2$. Line segment C$_1$ given by $y = 0$ and $0 ≤ x ≤ x_0$ and the line segment C$...
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uniform Effect of K-means Clustering

In the following link is discussed the uniform Effect of K-means Clustering: https://www.springer.com/cda/content/document/cda_downloaddocument/9783642298066-c2.pdf?SGWID=0-0-45-1338325-...
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1answer
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Is this vectorial identity between operators true?

$$\nabla^2(\mathbf{t} u(\mathbf{r}) ) =\mathbf{t} div(\nabla u(\mathbf{r}))$$ Where $\mathbf{t}$ is constant vector and $\nabla^2$ is the vector laplacian (defined here: https://en.wikipedia.org/wiki/...
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2answers
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Multivariable function equivalent definition

Say a function $f: B(x;r) \rightarrow \mathbb{R}^q$ is a continuously differentiable function with $\|Jf(x)\| \leq c$ for all $x \in B(x;r)$. I want to show that $\|f(x_1) - f(x_2)\| \leq c\|x_1-x_2\|$...
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Why do we define divergence of a vector field as $\lim\limits_{\delta V\to 0}\frac1{\delta V}\oint_{s} \vec v\cdot d\vec A$

Why do we define divergence of a vector field as $\lim\limits_{\delta V\to 0}\frac1{\delta V}\oint_{s} \vec v\cdot d\vec A$ I think it is because of the Divergence Theorem but this formula comes to ...
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1answer
26 views

Octahedron Pyramid

So, each octahedron can be inscribed in a cube, so that the corner points of the octahedron are in the midpoints of the side areas of the cube, am I right? From the octahedron $ABCDS_1S_2$, shown in ...
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1answer
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What is the proof for the Vortex vector field equation?

I'm Struggling to understand why the vortex vector field is given by: Vortex vector field equation $\vec F(x,y) = (\frac{-y}{x^2+y^2}, \frac{x}{x^2+y^2})$ If anyone could explain why this is, I ...
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Test the following identities of the vector analysis, for fields $F, G: \mathbb{R}^3→ \mathbb{R}^3$ and functions $f, g: \mathbb{R}^3 → \mathbb{R}$.

Test the following identities of the vector analysis, for fields $F, G: \mathbb{R}^3→ \mathbb{R}^3$ and functions $f, g: \mathbb{R}^3 → \mathbb{R}$. $\text{a)}\space\nabla \cdot (\nabla f \times \...
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2answers
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Converting a 2-form on $\mathbb{R}^{3}$ to a vector field on $\mathbb{R}^{3}$

How do I convert a 2-form $x dy \wedge dz + y^2 dx \wedge dz$ on $\mathbb{R^{3}}$ to a vector field on $\mathbb{R^{3}}$? Attempt: Suppose we have two vectors fields a and b in $\mathbb{R^{3}}$ such ...
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1answer
24 views

How do I properly set up this integral equation?

Using the gaussian law I want to determine the field strength of a charged stick on another load. The base formula for that is $$\epsilon_0 \oint {\vec{E} \cdot d \vec{A}} = q$$ A second formula for ...
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0answers
36 views

vector field: changing vector magnitudes to make it conservative

Consider a vector field $$\vec{F}(x,y)=P(x,y)\vec{i}+Q(x,y)\vec{j}$$ on an open and simply-connected region. Assume $P$ and $Q$ have continuous partial derivatives. Under which conditions there ...
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1answer
45 views

Verify divergence theorem of paraboloid beneath a plane

Verify the divergence theorem for the vector field $\mathbf F =3x^2y^2\mathbf i +y\mathbf j -6xy^2z\mathbf k$ for the volume bounded by the paraboloid $z=x^2+y^2$ and $z=2y$ . I tried to compute ...