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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-1
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1answer
22 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 $\vec{a}$: $\frac{\partial }{\partial \vec{a}} f(\vec{a},\vec{b})$, ...
0
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1answer
33 views

Help with exercise on vector field

Let's consider a vector field $v$ in a bounded region $R$ of the space; assume that $div\,v=0$ and $v=0\,\,\text{on}\,\, \partial R$; I have to prove that $$(\nabla v)^Tn=0$$ where $n$ is the outward ...
1
vote
1answer
59 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 ...
1
vote
1answer
23 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 ...
0
votes
2answers
52 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 ...
1
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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 ...
1
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0answers
27 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
57 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) ...
0
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1answer
45 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 ...
0
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2answers
29 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
votes
3answers
530 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
48 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
38 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} ...
2
votes
1answer
52 views

curve integral - intersection between plane and sphere

I am going to calculate the line integral $$ \int_\gamma z^4dx+x^2dy+y^8dz,$$ where $\gamma$ is the intersection of the plane $y+z=1$ with the sphere $x^2+y^2+z^2=1$, $x \geq 0$, with the orientation ...
0
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5answers
40 views

Prove that the gradient of a unit vector equals 2/magnitude of the vector

Let $\vec r=(x,y,z)$ Firstly find $\vec \nabla (\frac 1 r)$ where r is the magnitude of $\vec r$. I think I've done this correctly to get $-x(x^2+y^2+z^2)^{-\frac32} \hat i-y(x^2+y^2+z^2)^{-\frac32} ...
0
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1answer
22 views

Simple vector calc question

How would I approach the question: A volume $V$ is enclosed by a closed surface $S$. Show that $$\iiint_V \frac{1}{r^2} dV = \iint_S \frac{\underline{r}.d\underline{S}}{r^2} $$ where ...
0
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3answers
53 views

Find eigenvector of the linear operator

Task is to find an eigenvector of the following linear operator: $f \to \int^{x}_{-x} f(t)dt$ in the linear span $\langle cos(x), sin(x), ...,cos(mx),sin(mx)\rangle$. I know how to find eigenvectors ...
2
votes
1answer
46 views

Prove equality of two vectors if they have equal divergence and equal curls

I have following question: Fields with equal divergence and equal curls $F_1$ and $F_2$ are two vectors fields, you may write them as $F_1 = M_1i+N_1j+P_1k$, $F_2 = M_2i+N_2j+P_2k$. Suppose that ...
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1answer
47 views

Find the divergence of the following vector fields

Consider an arbitrary vector field $F$ $$\eqalign{F&=F_1\hat{i}+F_2\hat{j}+F_3\hat{k}\\ &=F_{C_1}\hat{e}_\rho+F_{C_2}\hat{e}_{\phi}+F_{C_3}\hat{e}_{z}\\ ...
1
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1answer
43 views

vector analysis, area of surface

I am trying to solve this question: 'Obtain the surface area of part of the sphere $x^2+y^2+z^2=a^2$ contained within the cone $z \tan \alpha= \sqrt{x^2+y^2}$ where $0 \le \alpha \le \pi/2$. ...
2
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2answers
103 views

Verify Gauss’s Divergence Theorem

I have this assignment which we have not tackled and am getting mixed up in the divergence theorem tutorials like this one ...
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0answers
25 views

Conditions of the vector-calculus Green's theorem

While studying the Green's theorem, I think a lot about whether there is an abundance of the condition (now called $X$) of $X:\quad$ $P$ and $Q$ (in the Wikipedia article, $L$ and $M$) have ...
0
votes
1answer
192 views

Finding an area of a propeller using double integration? Attempted, please help! :( [duplicate]

The boundary of the internal hole is given by $r = a + b\cos(4α)$ where $a > b >0$. The external boundary of the propeller is given by $r = c + d\cos(3α)$ where $c - d > a + b$ and $d > ...
3
votes
1answer
38 views

cross product in cylindrical coordinates

Hi i know this is a really really simple question but it has me confused. I want to calculate the cross product of two vectors $$ \vec a \times \vec r. $$ The vectors are given by $$ \vec a= a\hat ...
2
votes
1answer
50 views

Plane parallel to two lines and goes through a point?

Find a plane that is parallel to both $$\frac{x-1}2 = y = z+1$$ and $$x=\frac{y+1}3=\frac{z-1}4$$ and goes through the origin. Is it possible to place it so that it is equidistant to both lines? ...
0
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1answer
59 views

verifying the divergence theorem for the region in $R^3$ bounded by the surfaces $z=1-x^2$, $y=0$, $y=1$ and the x-y plane

I am stuck on the following question. Vector calculus is not a forte of mine. Let V be the region in $R^3$ bounded by the surfaces $z=1-x^2$, $y=0$, $y=1$ and the x-y plane. S is the closed surface ...
1
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2answers
645 views

Area of a weird ellipse shape.

A propeller has the shape shown below. The boundary of the internal hole is given by $r = a + b\cos(4α)$ where $a > b >0$. The external boundary of the propeller is given by $r = c + d\cos(3α)$ ...
1
vote
1answer
44 views

Gauss Divergence Theorem Calculation help

I am having trouble getting my head around what exactly is required in this problem. Let $S$ be an arbitrary piecewise smooth, orientable, closed surface enclosing a region $\mathbb{R}^3$. Calculate ...
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0answers
16 views

Defining divergence of vector field

Curl is defined (in the plane) by imagining a wheel of radius $\epsilon$ placed in $(a,b)$. Denote the region enclosed by the boundary of the wheel with $D_\epsilon$. Let's suppose our vector field ...
3
votes
3answers
58 views

Find dimension of the intriguing vector space

We are given vector space of polynomials over $\mathbb R$ of two variables with powers not higher than 2013. Let's consider subspace $V$ which contains such polynomials $f$, so following holds for ...
3
votes
2answers
34 views

Gauss's theorem in 2d: how can it be expressed in differential forms?

How do we express the 2d version of Gauss's theorem in the language of differential forms? In 3d, I know it is $$d \left(Fdydz + Gdzdx + Hdxdy\right) = F_x + G_y + H_z dxdydz$$ so by Stokes' ...
1
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0answers
34 views

derivation of divergence from nabla operator

For a two dimensional orthogonal curvilinear coordinate system $(t_1, t_2)$, we have the position vector $r$, where $h_i = | \frac{\partial r}{\partial t_i} |$ are the scale factors and $a_i$ are the ...
0
votes
1answer
11 views

Find the surface integral of some ellipsoid?

I got Stokes theorem all warmed up for this one! $$\int_{S}\int(Curl(\vec{F}))d\vec{s}$$ (That means delta cross F or curl of F) Where S is the ellipsoid $x^2 + y^2 + 2z^2 = 16$ And $\vec{F} = ...
1
vote
1answer
17 views

Find the area bounded by the hypercycloid

Parametrization : $x = acos^3(t), y = asin^3(t)$ $a>0$ If you can solve it for me that would be awesome.:D If not, can you give me some hints? Tell me how to set it up and stuff. It's solveable ...
0
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0answers
30 views

Is it correct that $\frac{d\theta}{d\varphi'}=\left(\frac{d\varphi'}{d\theta}\right)^{-1}$?

Let $\theta$ be $k\times1$ and $\varphi$ be $k\times1$. Then $\frac{d\theta}{d\varphi'}$ is a $k\times k$ matrix with entries $\frac{d\theta_{i}}{d\varphi_{j}},\,\, i,j=1,...,k$. I was wondering if ...
0
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0answers
40 views

understanding Green's theorem Intuition

The idea of it is to find the area of a region, yet I keep seeing vector fields popping up all over the place. Take this example from my text book: Find the region enclosed by the two graphs: $y = ...
4
votes
2answers
87 views

Why does Stokes theorem apply to this situation?

I'm thinking Green's theorem or stokes theorem, but I don't know. It has been driving me crazy all day. Help me out here! And if you don't want to help because you know it's homework, give me some ...
0
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1answer
32 views

Find the area bounded by the hypocycloid?

I have the answer. The hypobloid has parametrization = $x = acos^3(t)$ $y = asin^3(t)$ The explanation is you take a vector field $F(x,y) = (0, x) which has curl 1 than it says the area is equal to: ...
0
votes
1answer
38 views

Can this line integral problem be solved with Stokes theorem?

I have a feeling it could, or with some other theorem. $F(x,y,z) = (2xyz + \sin x)i + (x^2z)j + (x^2y)k$ $$\int_{c} F.ds$$ where $c(t) = (\cos^5(t),\sin^3(t),t^4)$ I tried it in differential form ...
2
votes
0answers
39 views

Is there a way besides integration by parts to solve this integral?

$$\int_{0}^{2\pi} -10\cos^9(t)\sin^4(t)t^4\,dt$$ Maybe a formula for this form or something?
1
vote
1answer
31 views

Evaluating a line integral through a vector field in 3 dimensions.

Let $\mathbf{F}(x,y,z) = (2xyz + \sin x)\mathbf{i} + (x^2 z)\mathbf{j} + (x^2 y)\mathbf{k}$. Evaluate the integral of $\mathbf{F}$ along $c$, where $c(t) = (cos^5(t), sin^3(t), t^4)$, $t \in [0, ...
5
votes
1answer
50 views

Is there a vector field that is the complete opposite of a conservative one

Is there a three-dimensional vector field such that for every non-selfintersecting closed curve (that is not just one point, to avoid degenerate cases) the respective line-integral on the curve ...
1
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1answer
72 views

Line integral + Work

$F=(z-y)i+(x-z)j+(2y-x)k$ Let $C$ be a curve formed by an intersection of the plane $2x-z=0$ with the cylinder of elliptical cross section $x^2+(y^2)/9=1$, assuming $y$ is parametrized along $C$ via ...
0
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1answer
53 views

Prove: If $\int_{\phi}\omega = \int_{\psi}\omega$ whenever $\phi$ and $\psi$ have the same endpoints, then $\omega=df$

This is an exercise that I have been assigned for homework. I don't really know how to approach it though. I know that $\int_{\phi}\omega$ only depends on the endpoints $\phi(a)$ and $\phi(b)$ where ...
0
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2answers
37 views

Clarification: What does it mean when “$\phi$ and $\psi$ are two smooth curves in $U$ with the same beginning and end points”

"$\phi$ and $\psi$ are two smooth curves in $U$ with the same beginning and end points" Does this mean: (A) $\phi:[a,b]\to U$ and $\psi:[a,b]\to U$ (B) $\phi:[a,b]\to U$ and $\psi:[c,d]\to U$ s.t. ...
2
votes
1answer
35 views

integrals of vector fields that yield vectors, not scalars

When I tried to think of how I'd answer this question, I realized that never in my undergraduate curriculum was I asked to compute the surface or line integral of a vector field. I don't mean I've ...
1
vote
1answer
53 views

Integral/Vector calculus $\oint_{\partial S} u \vec \nabla v \cdot d \vec \lambda=\int_S (\vec \nabla u)\times (\vec \nabla v)\cdot d\vec S.$

I am trying to show that $$ \oint_{\partial S} u \vec \nabla v \cdot d \vec \lambda=\int_S (\vec \nabla u)\times (\vec \nabla v)\cdot d\vec S $$ using Levi Cevita notation methods only. The Levi ...
0
votes
1answer
18 views

Vectors and Forces

A box weighting 294N is sitting on a ramp. If the ramp is inclined at an angle of 25 degrees to the horizontal, and there is a 40N force of friction, calculate the amount of force that must be ...
0
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1answer
15 views

$w$ is a form, find $\alpha$ s.t $d \alpha=w$

$w$ is a form, find $\alpha$ s.t $d \alpha=w$ $w=(2y-4)dy \wedge dz+(y^2-2x)dz \wedge dx+(3-x-2yz)dx \wedge dy$ $w$ is a 2-form and $d \alpha=w$ so $\alpha$ is a 1-form s.t: $\alpha =Mdx+Ndy+Pdz$ ...
1
vote
1answer
26 views

Let $\alpha= f\,dx_1 \wedge \cdots\wedge dx_n$; where $f$ is continuous on $A$. Show that $\int_ \Phi \alpha =\int_ \Phi f$

Let $A \subset \mathbb{R}_k$ be a rectangle (or box), and let $\Phi:A\to\mathbb{R}_k$, be the identity mapping. Let $\alpha= f \, dx_1 \wedge \cdots \wedge dx_n$; where $f$ is continuous on $A$. ...