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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Help with integral from Boltzmann equation

I have a function $$g(x,v,t) = u(x,t)· v + θ(x,t)\frac{1}{2}(|v|^2 - 5)$$ where $g(x,v,t),\theta(x,t)$ are scalars and $u(x,t),v∈ \Bbb R^N$, $N=2,3$. I also have a matrix valued function $X=X(v)∈\Bbb ...
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17 views

Mean curvature submanifold

Consider $S^{N-1}$ the unit sphere and let us focus our attention on the cap $$ G=S^{N-1}\cap\{x_N>0\} $$ with boundary $\partial G= S^{N-2}\times\{0\}$: it is quite obvious to see that $G$ is a ...
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17 views

How to find critical points in a cubic function in two variables?

Given a cubic function $f$ in two variables $x$ and $y$ $$ f(x,y)=\sum_{i=0}^3 \sum_{j=0}^3 k_{i,j}x^i y^j, $$ I would like to find the points ($x,y$ pairs) where $\nabla f = \mathbf{0}$. Since $f$ ...
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7 views

I am going to work in SE(3) group, is vector sum approach applicable in this group?

I am working on control of mobile robot in 3d. I want to do vector sum for X and Y components, use this vector sum in control methodology and again convert resulting speeds and torques into their X ...
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1answer
33 views

Calculate the integral $\iint_D (y^2-x^2)^{xy} (x^2+y^2)dxdy$ on a certain region

Let $D$ be the region that's bounded by $xy=a, xy=b, y^2-x^2=1, y=x$ in the first quadrant. Calculate the integral $\iint_D(y^2-x^2)^{xy}(x^2+y^2)dxdy$. Firstly, I was able to show that the boundary ...
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21 views

Is $\sigma_u \times \sigma_v \neq \vec{0}$ essential for $\int_{\alpha} \vec{F} \cdot d\vec{r} = \iint_\sigma \mathrm{curl}\,\vec{F} \cdot d\vec{S}$?

I think I have proved the following version of Stokes' theorem: Teorem 1: Let $\beta: [0,4] \to \mathbb{R}^2$ be the curve given by \begin{equation} \beta(t) = \begin{cases} (t,0) & \mbox{if ...
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2answers
18 views

Parabola, tangent and angles (Apostol, chapter 14.21, problem 1)

Apostol, chapter 14.21, problem 1 (a review problem) Here is the question: Let r denote the vector from the origin to an arbitrary point on the parabola $y^2 = x$, let $\alpha$ be the angle that ...
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25 views

Apply flow of $V$ to a segment of a curve, Do you get covariant derivative?

Apply flow of $V$ to a segment of a curve, look at the velocity of the resulting new curve, a perturbation of the original curve. How is the new velocity (tangent) related to the original one (right ...
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1answer
27 views

Difference of inner product space of two vectors

If in an inner product space $\alpha,\beta$ are two vectors such that $\|\alpha\|= 2,\|\beta\|=3$, and $\|\alpha+\beta\|=5$. Then $\|\alpha-\beta\|$ is equal to ? The options are 1)0 2)1 3)√10 ...
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1answer
21 views

let F be velocity vector field of fluid on $R^3$ defined by F(x,y,z)=-yi+xj.

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Vec}[1]{\mathbf{#1}}$Let $F$ be velocity vector field of fluid on $\Reals^3$ defined by $F(x,y,z) = -y\Vec{i} + x\Vec{j}$. (A) Show that $F$ is ...
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24 views

Derivative of Frobenius norm expressions

For an optimization problem using the L-BFGS algorithm, I am trying to use the gradients of two norm expressions. X are matrices, x elements of X. $$R_a = \Lambda * \sum_{c=1}^C ||X_c - 1/C ...
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29 views

Given u and v are two vectors, find ||u×v||² [closed]

If u and v are two vectors, then the value of ||u×v||² is equal to ?
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36 views

Stokes' theorem without the smoothness condition

I think I have proved the following version of Stokes' theorem: Teorem 1: Let $\beta: [0,4] \to \mathbb{R}^2$ the curve given by \begin{equation} \beta(t) = \begin{cases} (t,0) & \mbox{if } ...
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1answer
43 views

Find for which $\alpha$ $y=8x+\alpha$ is tangent to the curve $x^4+y^4=1$

Find for which $\alpha \in \mathbb{R}$, the line $y=8x+\alpha$ is tangent to the curve $x^4+y^4=1$. Firstly, I calculated the tangent to the curve, which is $(4x^3, 4y^3)$, and if the line is tangent ...
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1answer
42 views

Higher order terms in Hessian of $g(x)^T g(x)$, where $g(x)$ is the gradient of underlying $f(x)$: $\mathbb R^n \Rightarrow \mathbb R$

Consider a (continuously differentiable as many times as you need it) function $f(x)$: $\mathbb R^n \Rightarrow \mathbb R$. Let $g(x)$ = gradient of $f(x)$ w.r.t. $x$. Let $H(x)$ = Hessian of ...
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2answers
24 views

Meaning behind directional derivative

My task was to find the directional derivative of function: $$z = y^2 - \sin(xy)$$ at the point $(0, -1)$ in direction of vector $u = (-1, 10) $. The result I found was $-21/\sqrt{101}$. But I ...
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1answer
44 views

Gröbner basis is not a vector basis?

We use the same notation for Gröbner basis and vector basis. I recall that $\langle 1\rangle_{GR}$ is the largest Gröbner basis while $\langle 1\rangle_{vector}$ is the smallest vector basis. So for ...
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34 views

Basis of all real polynomials?

I am studying the book Topics in Algebraic Graph Theory by Beineke et all and the page 12. By the book, the set of all real polynomials can be generated by the set $\{1,x,x^2,\ldots\}$ which I ...
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1answer
51 views

Prove it has a definite solution.

I'm stuck on this problem. I don't even know how to start: If $(f_1,...,f_n) : \mathbb{R}^n\longrightarrow{\mathbb{R}^n}$ with $f \in C^1 $ is a vector field and ...
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1answer
31 views

Question related to differentiable functions on Banach spaces

There is an interesting exercise on my Analysis book that I have not been able to solve: Let $\mathbb{E,F}$ be Banach spaces, $f:\mathbb{E}\to\mathbb{F}$ of type $\mathcal{C}^k$, $k\geq1$. Asume ...
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1answer
43 views

What is the cross product integral?

I understand the integral $\int \vec{u} \cdot \mathrm{d} \vec{v}$ is a line integral but what is the integral $\int \vec{u} \times \mathrm{d} \vec{v}$ and how does it work? For example, how would I ...
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2answers
42 views

What kinds of motion obeys a higher order form of angular motion?

Angular velocity $\vec{\omega}$ can be defined in terms of velocity $\vec{v}$ and position $\vec{s}$ as: $$ \vec{\omega} = \frac{\vec{s} \times \vec{v}}{\left\lvert s\right\rvert^2} $$ Constant ...
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4answers
116 views

Parametrizing the surface $z=\log(x^2+y^2)$

Let $S$ be the surface given by $$z = \log(x^2+y^2),$$ with $1\leq x^2+y^2\leq5$. Find the surface area of $S$. I'm thinking the approach should be $$A(s) = \iint_D \ |\textbf{T}_u\times ...
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1answer
18 views

To calculate the flux of water through a parabolic cylinder

If velocity vector is given as $\mathbf F=y\mathbf i +2 \mathbf j+\mathbf k$ , then find the flux of water through the parabolic cylinder $y=x^2$, $0\le x\le 3$, $0\le z \le 3$. For this ...
3
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1answer
27 views

Suppose that the height of a hill above sea

Suppose that the height of a hill above sea level is given by $z=8x^2y^3+x^3y+6$. If you are at the point (2,1,46) in what direction is the elevation changing fastest? What i have done? The ...
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12 views

Apply gradient and divergence on 2D matrix to get drains

I need to find all drains in relief map. It is 2D matrix with heights for each point. Is it possible to compute vector field from scalar 2D matrix by applying gradient and to compute scalar field of ...
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1answer
25 views

How to find the Surface Area?

Find the surface area of the paraboloid $z = x^2+y^2$ that lies inside of the cylinder $x^2+y^2 = 4$. I keep getting $\frac{\pi}{6} (17\sqrt{17} - 1)$ This is how I did it:
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28 views

Confusing moment from Theorem 10.43 from Rudin PMA

Point $(b')$ little bit confuses me. Let $\nabla \cdot \mathbf{F}=0$ then $d\omega_{\mathbf{F}}=0$. We see that $\mathbf{F}\in C'$. Since $\omega_{\mathbf{F}}$ is closed in $E$ then by theorem 10.40 ...
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10 views

Vector Identity in fem strong to weak form

Hi all, Can anyone please help me understand (step by step) how the first term in the last relation appears? (these steps was to convert the first relation from the strong form to the weak form in ...
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10 views

How to derive the procedure for scatter matrix

I am studying about the multiple discriminant analysis and I am suffer from the matrix calculataion. I think it is so easy, but it is not easy for me I am welcome all of you hints and comments and ...
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1answer
29 views

What's the Jacobian of the sign function for vectors?

What's the Jacobian of the sign function for vectors or: $$A = \frac{\mathrm{d} \hat{v}}{\mathrm{d} \vec{v}}$$ I think it is probably some kind of dirac delta or something like: $$A\vec{u} = ...
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1answer
18 views

Parallelism of Vectors

I know that two lines are parallel if they never intersect each other. The conditions for parallel vectors says that if a and b are two vectors then they are parallel if a=kb for k being a scalar. ...
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[svm]A Problem of max(1/|w|) equal to min(1/2*|w|^2)

I've been search many SVM theory thesis for machine learning Those articles usually say max(1/|w|) equal to min(1/2*|w|^2) but they didn't write the detail of the mathematics process. I also read this ...
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1answer
52 views

Differentiating the pull-back of a one-form

Let $\Omega$ be an open subset of a vector space $V$ and let $\alpha\colon \Omega\to V^*$ be a one-form on $\Omega$. Assuming that $\alpha$ is differentiable, then for any $x\in \Omega$, $D\alpha ...
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22 views

Proving the invariance of the mixed product under direct congruent transformations

Text: Differential Geometry, by Erwin Kreyszig Given a set of vectors: $$ \overrightarrow{a}=\langle a_1, a_2, a_3 \rangle $$ $$ \overrightarrow{b}=\langle b_1, b_2, b_3 \rangle $$ $$ ...
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2answers
27 views

Geometry of Vectors

I know the definition of collinear vectors and the condition for collinearity says "Two vectors $a$ and $b$ are collinear if $a=kb$, $k$ being non-zero scalar" but I am confused if $k=0$ then will not ...
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1answer
54 views

Poincare's lemma from PMA Rudin

It's the so called Poincare's theorem from Rudin's book. I read this theorem fully but I have 2 questions: 1) How he conclude that equations (120) holds? What did he use in his reasonings? This ...
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28 views

Is it possible to express the jacobi derivative of position with respect to angular displacement easier?

Let $$ \vec{\omega} = \frac{\vec{s} \times \vec{v}}{\left\lvert s\right\rvert^2} $$ and define $$ \vec{\theta} = \int \vec{\omega} \,\mathrm{d} t $$ $$ R = \frac{\mathrm{d} \vec{s}}{\mathrm{d} ...
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12 views

Is the inverse of the jacobian matrix the opposite derivative?

Is it the case that: $$\frac{\mathrm{d} \vec{u}}{\mathrm{d} \vec{v}}^{-1} = \frac{\mathrm{d} \vec{v}}{\mathrm{d} \vec{u}}$$ for the jacobian matrix barring zeros and noninvertible matrixes of ...
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1answer
23 views

Calculating Flux of F across G with paraboloid and plane

Calculate the flux of F across G where $\mathbf F(x, y, z) = 6x\mathbf i + 6y\mathbf j + 2\mathbf k$; G is the surface cut from the bottom of the paraboloid $z = x^2 + y^2$ by the plane z = 3 I found ...
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20 views

Showing that$(\vec{a}+\vec{b})\cdot[(\vec{b}+\vec{c})\times(\vec{c}+\vec{a})]=2\vec{a}\cdot(\vec{b}\times\vec{c})$ [duplicate]

$$(\vec{a}+\vec{b})\cdot[(\vec{b}+\vec{c})\times(\vec{c}+\vec{a})]=2\vec{a}\cdot(\vec{b}\times\vec{c})$$ Is there an option to prove it using the properties of dot and cross product? or do I need ...
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4answers
52 views

How does the chain rule work for functions from vectors to vectors?

Suppose I have a function: $$ \vec{s} = \vec{f}\left(\vec{\theta}\right)$$ and a derivative: $$ \vec{v} = \frac{\mathrm{d} \vec{s}}{\mathrm{d} t}$$ How do I apply the chain rule? For simplicity ...
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1answer
32 views

Using Euler's equation and vector identity

An unsteady incompressible inviscid fluid flow satisfies the continuity equation $∇·\textbf u = 0$ and Euler’s equation $$\frac{∂ \textbf u }{∂ t} +(\textbf u·∇)\textbf u = − \frac1ρ ∇p$$ where $\textbf ...
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vector analysis questions review

I am new to vector analysis and have attempted the questions below, I dont have any answers or person that can solve them for me. So if anyone could tell me wether my working and answers are correct ...
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2answers
35 views

What does it mean to integrate a vector function?

What is the meaning of a vector function and what is the geometric interpretation of integrating such a function?
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1answer
53 views

Prove $\oint_C\vec F \cdot \hat n\;ds=4\pi\,(q_1+…+q_n)$

Let $F:\Bbb R^2-\{p_1,p_2,\dots,p_n\} \to \Bbb R^2$, where $\{p_1,p_2,\dots,p_n\}\in \Bbb R^2$ be defined as $F(x)=\sum_{i=1}^n q_i \nabla\left(ln||x-p_i||^2\right)$ with $\{q_1,q_2,...,q_n\}\in \Bbb ...
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2answers
44 views

Gradient of vector field in spherical coordinates

I need to calculate the Hessian matrix of a scalar in spherical coordinates. To do so, I tried to determine the gradient of the gradient. Hence, I want a gradient of a vector field. My question is: ...
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1answer
18 views

Understanding tensor fields

This is just a simple question: Is every vector field a first-order tensor field? I understand the definition of vector field, but I have problems understanding first-order tensor fields.
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1answer
34 views

Circle as oriented $1$-simplex and it's boundary

Let $\gamma(t)=(r\cos t, r\sin t)$ where $r>0$ is fixed and $t\in [0,2\pi]$. Rudin write that it's an "oriented 1-simplex". Also he states that $$\partial \gamma=0.$$ Let $T(u)=(r\cos u,r\sin u)$ ...
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26 views

How do I indicate this identity?

Here,I want to show the following. \begin{align} I &=\int(\nabla\times\textbf{h})\cdot (\nabla\times\delta \textbf{h})d\textbf{r} \\ &= \int\nabla \times \nabla \times \textbf{h} \cdot\delta ...