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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64 views

$\nabla \times \nabla_u (u \cdot v)=0?$

Motivated by the vector identity $\nabla \times \nabla u=0,$ I would like to ask if this is also true for the gradient w.r.t. $u.$ $\nabla_u$ is defined by $$ \nabla_u(u \cdot v)=v \cdot \nabla ...
2
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1answer
63 views

Need help understanding a certain vector integral identity (Stokes' theorem corollary)

The Vector Integral page on the Wolfram mathworld website lists as eq.(4) the following vector integral identity: If $$\mathbf{F}:=\mathbf{c}\,F,$$ then ...
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1answer
55 views

Finding loci of possible points satisfying vector simultaneous equations

I recently had an exam and a question came up which I was only partially able to answer. The question was the following: Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be constant vectors in ...
2
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4answers
64 views

Negative Volume using $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})$

So, my textbook explains how to find the volume of a paralelpiped using $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})$. Makes sense, basically. But, when I go to do problems some combinations ...
2
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1answer
80 views

Spivak's “Calculus in Manifolds” problems

I have some troubles with this problems. Problem 1.18: If $A \subset [0,1]$ is the union of open intervals $(a_i,b_i)$ such that every rational number of $(0,1)$ is contained in $(a_i,b_i)$, for ...
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0answers
49 views

Stone-Weierstrass Approximation Theorem for Vector-Valued Functions?

A classical result in analysis is the Stone-Weierstrass approximation theorem: Theorem. Let $X$ be a compact Hausdorff topological space and $(C(X,\mathbb C),\|\cdot\|_{\infty})$ denote the space ...
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1answer
78 views

Stoke's Theorem Example - Help?

From Stoke's Theorem: \begin{equation*} \oint_c \textbf{F}\centerdot d\textbf{r}= \int\int_S (\nabla \times\textbf{F})\centerdot d \textbf{S}\end{equation*} Evaluate $\oint _C \textbf{F}\centerdot ...
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0answers
23 views

Vector Surface Integral problem finding ds

Question: Evaluate $∫F$.Nds where $F = 2x^2y \hat{\imath} -y^2 \hat{\imath} + 4xz^2 \hat{k}$ and $S$ is the closed surface of the region in the first ocant bounded by the cylinder $y^2+z^2 = 9$ and ...
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0answers
65 views

Calculating Eigenvector Centrality & Betweenness Centrality formulas explained in simple terms

I'm currently working on a software application that has a function that analyses networks of people and the relationships between them. Two of the important variables we look at are Eigenvector ...
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1answer
41 views

How to find the projection of a cylinder projected onto a plane?

Say you are given the equations: $x + y + z = 6$ and $x^2 + y^2 = 1$ You can easily find the plane and cylinder accordingly. But how do you find the projection of the cylinder onto that plane. The ...
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2answers
32 views

Finding a vector equation

I need to find the directional derivative of a vector. However the direction of the line is given in the form $(x-1)/2 = (y-3)/-2 = z$ form. How do I find the vector equation for the direction from ...
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1answer
28 views

Calculus of vector functions: arc length and speed

Hi! I am trying to study or a test in my calc3 class by doing some online problems, but I am not quite sure how to solve this one. I thought the correct answer was ...
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1answer
20 views

Is there a function to tell if two probability vectors' max values are in the same dimension?

Is there a method or function to tell two probability vectors' max values are in the same dimension? Or Is there a bound for the angle of two normalized probability vector which their max values are ...
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1answer
72 views

Conceptual Question on Dot product

Orthogonal unit vectors If $u_1$ and $u_2$ are orthogonal unit vectors and v = $au_1$ + $bu_2$ (a and b being scalars), find $v\cdot u_1$ . I know that by the dot product properties, $u_1 \cdot ...
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0answers
32 views

Gradient field and flow across a curve.

Say we have a gradient field $\vec{F}(x, y) = \nabla \phi$ and a closed curve $C$. We know that the curl $\nabla \times \vec{F}(x,y)$ will be $0$ and the net flow of this field along the closed curve ...
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1answer
99 views

Finding out whether normal points outwards or inwards

Consider $F(x,y,z)=(x,x^2y,0)$ and $$\Omega=\{(x,y,z)\in\mathbb{R}^3\mid(x^2+y^2)^2<z<\sqrt{x^2+y^2}\}$$ I want to compute $\iint_{\partial\Omega}(F\cdot\nu)\text{ds}$ where $\nu$ is the normal ...
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1answer
33 views

Integration of a vector function through a cilinder

I need to integrate $\vec F(x,y,z) = (2x,-3y,z)$ through a surface $\Sigma$. $\Sigma$ is defined as the surface of the cylinder $x^2+y^2=1$ (so without the top and the bottom) that is confined between ...
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3answers
57 views

Show that no application $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, of $C^k$ class, $k \geq 1$ can be injective

How can I proof this: Show that no application $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, of $C^k$ class, $k \geq 1$ can be injective, i.e., there are $A,B \in \mathbb{R}^2$ such that $A \neq B$ ...
2
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1answer
51 views

Line and Surface Integrals

I am stuck on the following question :( $F(x,y,z)=(y+z)i+(x+z)j+(x+z)k$. The sphere $x^2+y^2+z^2=a^2$ intersects the postive x−, y−, and z−axes at points A, B, and C, respectively. The simple closed ...
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20 views

Calculate the surface integral with Divergence theorem

Calculate $$\iint_YF*NdS$$ there $Y$ is the part of the surface $(x-z)^2+(y-z)^2=z $ there $z \leq1$. I cannot use Gauss theorem now since the boudnary of the volume V in the tripple integral ...
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1answer
29 views

A case of divergence theorem

$$\hat{n}_2=-\hat{n}_1$$ $$\iiint_D \nabla \overrightarrow{F} dV=\iiint_{D_1} \nabla \overrightarrow{F} dV+\iiint_{D_2} \nabla \overrightarrow{F} dV$$ $$\\$$ $$\iint_S \overrightarrow{F} \cdot ...
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4answers
70 views

Which is the normal vector??

Apply the divergence theorem over the region $1 \leq x^2+y^2+z^2 \leq 4$ for the vector field $\overrightarrow{F}=-\frac{\hat{i}x+\hat{j}y+\hat{k}z}{p^3}$, where $p=(x^2+y^2+z^2)^\frac{1}{2}$. $$$$ ...
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1answer
44 views

Variant on divergence theorem

If I want to prove that for any scalar field $f:\;\mathbb{R}^3\to\mathbb{R}:$ $$\int_V \boldsymbol{\nabla} f\;\mathrm{d}V=\int_{\partial V} f\;\mathrm{d}\mathbf{S}$$ Can I apply the divergence theorem ...
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0answers
49 views

Questions about the divergence theorem

I am looking at the proof of the divergence theorem and I have some questions. The proof of the divergence theorem $$\iiint_D \nabla \cdot \overrightarrow{F} dV= \iint_S \overrightarrow{F} \cdot ...
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0answers
30 views

What vector theorem should be used?

I have to integrate this line integral $$\int_C \mathbf F \cdot d\mathbf r$$ Where $\mathbf F = (\frac{y}{x^2 + y^2},\frac{-x}{x^2 + y^2})$ and $C$ is the curve $x^2 + 2y^2 = 1$ oriented ...
3
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1answer
113 views

What is the meaning of $dA$ in double integrals?

What is the meaning of $dA$ in $\iint_E\dots dA$, where $E$ is a region in the $xy$ plane? In some integrals we use $dA=dx\,dy$, but in others $dA=\hat{k}\,dx\,dy$. (Here $\hat {k}$ is the unit ...
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1answer
42 views

Is it known that $M=\frac{-y}{x^2+y^2} \text{ and } N=\frac{x}{x^2+y^2}$,when we have a circle?

There is the following graph in my notes: and then there is the formula: $$I=\oint _C \frac{xdy-ydx}{x^2+y^2}=2 \pi$$ So, at the formula of the Green theorem: $\displaystyle{M=\frac{-y}{x^2+y^2}} ...
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0answers
49 views

Line integral - Stokes's theorem

Let $\gamma_1$ be the intersection curve between the surface $x^2+(y+z)+z^2=1$ and the plane $z=x$ and let $\gamma_2$ be $\gamma_1$ with $x\geq0$,$y\geq0$. Calculate ...
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0answers
40 views

Proof of Green's Theorem

I am looking at the proof of the Green theorem. To show that $$\oint _S (Mdx+Ndy)= \iint_R \left( \frac{\partial{N}}{\partial{x}}-\frac{\partial{M}}{\partial{y}} \right) dxdy$$we do the following: ...
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0answers
17 views

Why is there a minus sign ($-$) , before the integral $\int_{S_2} M dx$?

I am looking at the proof of the Green theorem. $$S_1: y_1=f_1(x), a \leq x \leq b$$ $$S_2: y_2=f_2(x), a \leq x \leq b$$ $$\iint_R \frac{\partial{M}}{\partial{y}}=\int_{f_1(x)}^{f_2(x)} \int_a^b ...
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1answer
21 views

Is it also possible that the line intersects the curve at one point?

I am looking at the Green theorem: $$\text{ Let S be a simple closed curve of the plane xy,}\\ \text{such that a line that is parallel to each of the axes intersects the curve S at , at most, two ...
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2answers
63 views

Prove dot product identity $u\cdot v = \tfrac14(|u \cdot v|^2 − |u − v|^2).$

$$u\cdot v = \dfrac14\left(|u \cdot v|^2 − |u − v|^2\right).$$ So far I've only gotten the RHS to $\tfrac14((u \cdot v)(u \cdot v) − |u|^2 + 2(u \cdot v) - |v|^2)$ Only way I see this working is if ...
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1answer
37 views

Calculating the mass flux through the curve $AB$

Flux through a flat curve We want to calculate the mass flux through the curve $AB$ $$\Delta m= \delta \cdot \Delta s \cdot \Delta t \cdot \overrightarrow{v} \cdot \hat{n}$$ ...
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1answer
13 views

do we have to take at the beggining the variable $t$ and then $s$ , is there a difference?

I am given the following exercise: Calculate the work for the force if $$\overrightarrow{F}=\overrightarrow{i}(x^2-y)+\overrightarrow{j}(y^2-z)+\overrightarrow{k}(z^2-x)$$ where the path of ...
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1answer
27 views

What does $\hat{T}$ represent?

I am looking at line integrals and work.. According to my notes: $$\frac{\overrightarrow{R}}{dt}=\hat{T} \cdot |\frac{d\overrightarrow{R}}{dt}|=\hat{T} \sqrt{x'(t)^2+y'(t)^2+z'(t)^2}$$ So,the work ...
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2answers
42 views

Integrate the function $w=x+y^2$

I have the following exercise: We want to integrate the function $w=x+y^2$ and we have a path that begins from $A(0,0)$ and reaches at $B(1,1)$. $$$$ Could you give me some hint what I am supposed ...
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2answers
34 views

Proof of vector norm formula

I was trying to find something that proves the (Euclidean) vector magnitude formula for $3+$ dimensional vectors. $$\mid x|=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}.$$ It seems easy to prove by induction if ...
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1answer
25 views

derivative of sum of vectors

suppose i need to make the partial derivative of this vector function $f(\vec{a},\vec{b})=\frac{1}{| \vec{a}+\vec{b}|}$ respect to $\vec{a}$: $\frac{\partial }{\partial \vec{a}} f(\vec{a},\vec{b})$, ...
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1answer
37 views

If a vector field with zero divergence vanishes on a flat portion of boundary, its normal derivative is zero

Let's consider a vector field $v$ in a bounded region $R$ of the space; assume that $\operatorname{div}v=0$ and $v=0\,\,\text{on}\,\, \partial R$; I have to prove that $$(\nabla v)^Tn=0$$ where $n$ is ...
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1answer
60 views

$(n\times \bigtriangledown \times n)^2$ Specific equation?

The problem i'm trying to solve is $(\mathbf{n \times \bigtriangledown \times n})^2$ $\mathbf{n \times \bigtriangledown \times n = n_{30}}$ where ...
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1answer
24 views

(nx(gradxn))^2 operator question?

by $A\times B \times C = (A \cdot C)B-(A \cdot B)C$, i need to expand $n \times \bigtriangledown \times n$, where all of these are vectors. Here is what i have right now $n \times \bigtriangledown ...
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2answers
57 views

Integration over the cube

I have the following exercise: Integrate the $g=x \cdot y \cdot z$ over the cube that is on the first octant and that is bounded from the levels $x=1, y=1, z=1$. Could you give me some hint what I ...
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1answer
51 views

Questions about the line integral

Here's how we get to the formula for the line integral: $$\overrightarrow{R}(t)=x(t) \hat{\imath}+y(t) \hat{\jmath}+z(t) \hat{k}, \ \ \ \ \ \ a \leq t \leq b$$ We subdivide the curve into the ...
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0answers
58 views

How to estimate/determine surface normals and tangent planes at points of a depth image (point cloud)?

I have depth image, that I've generated using 3D CAD data. This depth image can also be taken from a depth imaging sensor such as Kinect or a stereo camera. So basically it is depth map of points ...
2
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1answer
60 views

Questions about the surface integral

Here's how we get to the formula for the surface integral: $$\Delta P_k=\frac{\Delta A_k}{\cos{\gamma}}$$ $$g:\text{ density }$$ $$\text{ Integral }=\sum_k \Delta P_k \cdot g(x_k, y_k, z_k) ...
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1answer
48 views

We consider a fluid which rotates around the $z-$ axis with constant angular speed $w$

I have the following exercise in my notes and I need some explanations. We consider a fluid which rotates around the $z-$ axis with constant angular speed $w$ $$\overrightarrow{r}=x \hat{\imath}+y ...
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2answers
31 views

Definition of vector field [closed]

How would you explain the definition of a vector field? I have found the following definition: If at a point $P$ of a region $G$ is given a vector $F(P)$ the set of all these vectors is a called ...
3
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3answers
566 views

How to prove that $\nabla (a\cdot b)=(a\cdot \nabla )b+(b\cdot \nabla )a+a\times (\nabla \times b)+b\times (\nabla \times a)$?

I'm working hard to prove this.. $$\nabla (a\cdot b)=(a\cdot \nabla )b+(b\cdot \nabla )a+a\times (\nabla \times b)+b\times (\nabla \times a)$$ but I got $$\nabla (a\cdot b)=\nabla (a\cdot b)+\nabla ...
0
votes
1answer
54 views

Vector Calculus Identity help

I am having some issues with the following question: Prove the following vector calculus identity in $\mathbb{R}^3$, where $f$ is a twice continuously differentiable scalar field and $F$ is a twice ...
0
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2answers
39 views

What is divergence of this function?

Let $$\vec{f}(x_1,x_2) = g_1(x_1,x_2) \hat{i} + g_2(x_1,x_2) \hat{j} + g_3(x_1,x_2) \hat{k}$$ then using the definition of divergence we get, $$\mathrm{div} f = \sum_{i = 1}^{2} ...