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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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 ...
0
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0answers
39 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
15 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 ...
0
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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
57 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 ...
1
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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 $AB$ $$\Delta m= \delta \cdot \Delta s \cdot \Delta t \cdot \overrightarrow{v} \cdot \hat{n}$$ ...
0
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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 ...
0
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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 ...
1
vote
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
33 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 ...
-1
votes
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 $\vec{a}$: $\frac{\partial }{\partial \vec{a}} f(\vec{a},\vec{b})$, ...
0
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1answer
34 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
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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
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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 ...
0
votes
2answers
56 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
45 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
votes
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) ...
0
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1answer
47 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
votes
3answers
549 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
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1answer
51 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} ...
2
votes
1answer
58 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
votes
5answers
47 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
votes
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
votes
3answers
63 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
53 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 ...
1
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1answer
54 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
vote
1answer
46 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
112 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
29 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
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1answer
194 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
105 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
59 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
63 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
vote
2answers
648 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
45 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
38 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' ...
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0answers
42 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
18 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
31 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
votes
0answers
48 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
89 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 ...
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1answer
34 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
40 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?