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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0answers
31 views

Jacobian matrix of the parametrization of (part of) a ball

I read (in E. Sernesi, Geometria 2) that the function $\varphi:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\times\mathbb{R}^{n}\to\mathbb{R}^{n+1}$ defined by $$\varphi(\theta_1,\ldots,\theta_n,r)=\left(...
3
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2answers
63 views

Evaluate $\int_c \mathbf F\bullet d\mathbf r$ along a given path

I've been stuck on one of our practise problems for Engineering Mathematics all day and am unsure how to tackle this problem. The problem asks to evaluate $\int_C \mathbf F\cdot d\mathbf r$ along a ...
0
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1answer
32 views

Finding the scalar potential of a vector-valued function

I am trying to find the scalar potential of a particular vector-valued function that I am working with, and I find that the function in question is $$\mathbf{g}(x,y) = \left( \frac{2x+y}{1+x^2+xy} , \...
1
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1answer
46 views

gradient of trace$(ABA^TC)$ w.r.t a Matrix A.

With n-order Matrix A,B,C.I was trying to find $ \nabla_A trace(ABA^TC)$ This answer:Proof for the funky trace derivative : $d (\operatorname{trace} (ABA'C))$? suggested: $$ \nabla_A \...
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1answer
22 views

Can a point and a direction vector represent a line?

I understood that to represent a plane all that is needed is a point a direction vector. But here it represent a line, how could it be?
0
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0answers
54 views

Gradient descent in n-dimensional space in the context of an RBF network

I am trying to implement an algorithm to perform gradient descent in a $n$-dimensional space in the context of an RBF network. My network has 5 inputs and 1 output. It has the following Gaussian ...
0
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1answer
14 views

Help with normal vectors to linear vector functions.

I am studying basic vector calculus and am on tangential and normal vectors. I understand why the derivative of a vector is tangential, and I also understand why the second derivative of a vector ...
1
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2answers
35 views

Rayleigh quotient $Q=(\frac{||\triangledown w||}{||w||})^2$ in using the eigenfunction $\sin(x)$ on the segment $(0,\pi)$

I would like to well understanding the Rayleigh quotient $Q=(\frac{\|\nabla w\|}{\|w\|})^2$. Does anyone could explain to me why we divide the norm of the gradient $\| \nabla w \|$ by $\| w \|$, and ...
0
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0answers
22 views

Integrating vectors

If $\displaystyle \int_a^b \mathbf{F}\cdot\mathrm{d}\mathbf{v} = \int_a^b [F_x,\,F_y,\,F_z]\cdot\mathrm{d}[x,y,z] = \int_a^b F_x\mathrm{d}x+F_y\mathrm{d}y+F_z\mathrm{d}z$ isn't it then easier to ...
0
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0answers
18 views

Gradient 2 variables (2D or 3D)

Let's assume an example with 2 variables (mountain): http://math.arizona.edu/~calc/Surface.JPG: My question: The gradient points in the direction of greatest increase; Let's assume you stand on the ...
-1
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0answers
17 views

Integral of a $C^{\infty}$ vector field is independent from the parameter

Given a vector field $F \in C^{\infty}$, let $\Sigma_a:=\{ (x,y,z) \in \mathbb{R}^3 | z=a(1-(x^2+y^2)), x^2+y^2 \leq 1 \}$ be a surface and let $n=(n_i)_{i=1,2,3}$ be the normal unit vector s.t. $...
1
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1answer
26 views

Find the absolute max and min of a multivariable function on a bounded by a circle?

So i do understand everything up the square rectangle, in the photo here i mean, how did he come up with $(±2,0), (0,±1)$ is it because of $g(2cos x, sin x)$ and if that is the case why would he ...
0
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0answers
42 views

How do I interpret this dirac delta integral?

In this paper (in the appendix, page 16) I found the following integral, $$∭_{\Bbb S^2\times \Bbb S^2\times \Bbb S^2 } \delta( r\sigma + r_*\sigma_* - r'\sigma' - r'_*\sigma'_*) \ \text{d}\sigma_* \...
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0answers
11 views

Vector Calculcus identities involving second order tensors

I was reading a paper on Vortex Dynamics and came across the expressions $$\nabla(\vec{A}\cdot\vec{B})=(\nabla\vec{A})^T\cdot\vec{B}+(\nabla\vec{B})^T\cdot\vec{A}$$ and $$\vec{U_{b}}\cdot(\nabla\vec{a}...
1
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1answer
9 views

Is $\vec{F} (x,y,z) = (x^2(y - z)+ yz)\vec{i} + (2xyz + x/z)\vec{j} + (y/x - 2xyz)\vec{k}$ solenoidal ?

Check if div$\vec{F}$ = P′x+Q′y+R′z = or $\neq$ 0. Where $P = x^2(y - z)+ yz$, Q = 2xyz + x/z, R = y/x - 2xyz . Just as simple as that?
5
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0answers
134 views

Help with integral from Boltzmann equation

I have a function $$ g\left(x,v,t\right) = u\left(x,t\right)\cdot v + \theta\left(x,t\right)\frac{1}{2}\left(\left\lvert v\right\rvert^2 - 5\right) $$ where $g(x,v,t),\theta(x,t)$ are scalars and $u(...
1
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0answers
21 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 ...
0
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0answers
19 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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0answers
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 ...
1
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1answer
39 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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0answers
25 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 }...
0
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2answers
25 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 ...
1
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0answers
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 ...
0
votes
1answer
32 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 4)√...
1
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1answer
22 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 rotational....
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0answers
26 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 \sum_{c=1}...
1
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0answers
38 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 } t\...
1
vote
1answer
45 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 ...
0
votes
1answer
58 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 $f(x)...
1
vote
2answers
26 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 can'...
1
vote
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 ...
0
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0answers
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 ...
0
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1answer
52 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 $V:\mathbb{R}^n\longrightarrow{\...
1
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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 $f'(...
1
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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 ...
1
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2answers
43 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 ...
3
votes
4answers
123 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 \textbf{T}...
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1answer
32 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 problem I ...
3
votes
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 ...
0
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1answer
26 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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0answers
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 ...
0
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0answers
12 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 the ...
0
votes
0answers
11 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 ...
0
votes
1answer
33 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} = 2\...
0
votes
1answer
20 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. ...
0
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0answers
18 views

[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 ...
3
votes
1answer
56 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 (x)\...
1
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0answers
25 views

Proving the invariance of the mixed product under direct congruent transformations

Text: Differential Geometry, by Erwin Kreyszig Given a set of vectors: $$ \overrightarrow{a}=\langle a_1, a_2, a_3 \rangle $$ $$ \overrightarrow{b}=\langle b_1, b_2, b_3 \rangle $$ $$ \overrightarrow{...
1
vote
2answers
28 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 ...
0
votes
1answer
62 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 did he conclude that equations (120) holds? What did he use in his reasoning? This ...