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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Finding the surface integral of a vector field

How many cubic metres of fluid cross the upper hemisphere $x^2+y^2+z^2=1$, $z\ge0$ per second if the velocity of the flow is $\mathbf{u}=\mathbf{i}+x\mathbf{j}+z\mathbf{k}$ metres per second. So I ...
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3answers
31 views

Parametrising a surface for a surface integral

How many cubic metres of fluid cross the upper hemisphere $x^{2}+y^{2}+z^{2}=1$, $z\ge0$ per second if the velocity of the flow is $\mathbf{u} = \mathbf{i}+x\mathbf{j}+z\mathbf{k}$ metres per second. ...
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1answer
28 views

parametrising a surface

I want to parametrise the surface $x^{2}+y^{2}=36$ to then calculate a surface integral however I'm not really sure how to parametrise this. Can we use $$\mathbf{r}(t) = (6\cos{t}, 6\sin{t}, 0).$$ I ...
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1answer
21 views

Calculate line integral without integrating

Without carrying out any integration, show that the line integral $$\int_{C} \nabla\phi\cdot \mathbf{dr} = 3$$ where $C$ is any smooth curve joining $(-1, 3, 9)$ to $(1, 6,-4)$ and $\phi = xyz$. I ...
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1answer
49 views

surface integrals parametrising

Find a parameterisation and compute $r_{\alpha},r_{\beta},r_{\alpha}$ x $r_{\beta}$ and the tangent plane at the point mentioned of the surface $$x^2+y^2-z^2=2y+2z$$ where $-1\leq z \leq 0$ and the ...
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1answer
39 views

If divergence is zero, is it necessarily a curl?

The divergence of the curl of a vector is zero. But, Any vector whose divergence is zero can be the curl of a vector field ?
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1answer
30 views

Find $F_u(1, 1)$ and $F_{u,v}(1, 1)$.

Need help on this.. Suppose that $F(u, v) = f(x(u, v), y(u, v))$, where $f$ is a function satisfying \begin{cases}f(1, 2) = 3\\f_x(1, 2) = 1\\f_y(1, 2) = −2\\f_{x,x}(1, 2) = 3\\f_{x,y}(1, 2) = ...
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1answer
36 views

Show that $xg_x(x, y) + yg_y(x, y) = 0$.

Need help with this. Suppose that $G(u, v)$ is a differentiable function of two variables and that $g(x, y) = G(x/y , y/x)$. Show that $xg_x(x, y) + yg_y(x, y) = 0$. Where $g_x(x,y)$ and $g_y(x,y)$ ...
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0answers
37 views

What is meaning of $FFT(\vec{E}(x,y ))$

What is the meaning and how one takes fourier transformation of vector that has spatial distrubution. Let say electric field (with transfer x, y distibution) with direction $$FFT(\vec{E}(x,y ))$$ ...
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0answers
23 views

Projection of Vectors v on w

Given $v = [3, -6, 2]$ and $w = [-1, 6, 5]$, find; $v \downarrow w$ $w \downarrow v$ What does the magnitude of $w \downarrow v$ depend on? What does the direction of $w \downarrow v$ depend on? ...
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1answer
18 views

$|V|=?$ if $2\vec{V}+(\vec{V}\times(\vec{i}+2\vec{j}))=2\vec{i}+\vec{k} $

I am given $$2\vec{V}+(\vec{V}\times(\vec{i}+2\vec{j})=2\vec{i}+\vec{k} $$ and I need to find $$|V|=?$$ , I took dot product of equation with $\vec{V}$ but got stuck in $$2V^2 = ...
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2answers
65 views

$\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are on the same plane

We want to show that if $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are on the same plane, then there are $A, B, C$ not all $0$ such that $A \vec a+B \vec b+C \vec c=\vec 0$. $$$$ ...
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1answer
22 views

Two parallel planes

When two planes have the same perpendicular vector $\overrightarrow{v}$, then they are parallel, right?? We have that the two planes $Ax+By+Cz+D_1=0$ and $Ax+By+Cz+D_2=0$ are parallel, since they ...
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1answer
48 views

Triangle in space

Using vector notation describe the triangle ( in space ) that has as vertices the origin and the endpoints of the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. $$$$ The solution is the ...
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1answer
49 views

Velocity vector

We suppose that a ship, that is at the position $(1, 0)$ of a nautical map (with the North at the positive direction $y$) and it "sees" a rock at the position $(2, 4)$, is directed to North and is ...
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2answers
36 views

Angular velocity

The angular velocity $\omega$ of rotation of a rigid body has the direction of the rotaion axis and magintude equal to the rotation rate in rad per second. The orientation of $\omega$ is determined by ...
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1answer
43 views

Torque of a force

The torque $M$ of a force $\overrightarrow{F}$ as for the point $O$ is defined as the product of the magnitude of the force $\overrightarrow{F}$ and the perpendicular distance of the point $O$ and the ...
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2answers
70 views

Do these vector fields span an integrable distribution?

Let $X_i$, $1 \le 1 \le n$, be smooth vector fields on the open set $U \subset \mathbb{R}^n$, which are linearly independent at each point. Set $[X_i, X_j] = \sum_{k=1}^n c_{ij}^kX_k$. Suppose that ...
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1answer
38 views

Show that it is the volume

A liquid flows through a flat surface with uniform vector velocity $\overrightarrow{v}$. Let $\overrightarrow{n}$ an unit vector perpendicular to the plane. Show that $\overrightarrow{v} \cdot ...
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0answers
48 views

proving the laplacian of a vector in cylindrical coordnates

I am proving the following identity for the laplacian of a vector $\vec{v}=<v_r,v_\theta,v_z>$ in cylindrical coordinates: $$\nabla^2 \vec{v}=\left( \frac{\partial^2 v_r}{\partial ...
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0answers
49 views

Derivative of angle between two vectors singularity!

I have been battling a problem of needing to know the derivative of the angle between two vectors, the vectors possibly being parallel at some points in time. I started off with: $$\bf A \dot \bf B = ...
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2answers
32 views

finding potential f

Compute $f$ so that $F=\nabla f$ for $$F(x,y,z)=(yz+x-y, xz-x+z, xy+y-z)$$ I need to see the method of finding this that is not using the theorem about the star shaped set. The method that is like ...
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2answers
55 views

Poincaré–Bendixson theorem

Does someone know a good reference for a proof of the Poincaré–Bendixson theorem using the language of vector fields?
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2answers
20 views

Is the work integral decomposable?

Work is defined as $W = \int_{\gamma} \vec F \cdot d\vec l$ which I think means $W = \int (F_x, F_y, F_z) \cdot (dx, dy, dz)$. So by the linearity of the integral, could we always decompose work into ...
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5answers
71 views

Find the equation of the plane knowing that it passes through 3 points

I have to find the equation of the plane that passes through $(0, 0, 0), (4, 0, -2), (0, 8, -6)$. I have done the following: The equation of the plane is of the form $$ax+by+cz+d=0$$ Since the ...
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2answers
32 views

Find equation of plane

I have to find the equation of the plane that is perpendicular to the line $\overline{l}(t)=(10, 0, 4)t+(6, -2, 2)$ and passes through $(10, -2, 0)$. We know that a plane that has a perpendicular ...
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6answers
68 views

Unit vectors that are orthogonal to vectors

I have to find all the unit vectors that are orthogonal to the vectors $\overrightarrow{a}=(2, -4, 3), \overrightarrow{b}=(-4, 8, -6)$ . I calculated that the cross product $\overrightarrow{a} ...
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3answers
45 views

Find the equation of the line

I have to find the line that passes through $(3, 1, -2)$ and intersects under right angle the line $x=-1+t, y=-2+t, z=-1+t$. (HINT: If $(x_0, y_0, z_0)$ the intersection point, find the ...
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1answer
39 views

making up your F (Green's theorem)

Use Green's Theorem to find the areas enclosed by curve $$C: 2x^2+3y^2=2y$$ in $\mathbb{R}^2$. Can someone tell me how to make up $F.\,\,\,$
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2answers
42 views

Parametric equation of a line

When we want to find the line that goes through two points $(-2, -2, -2)$ and $(2, -2, 4)$ we use the formula $$\overrightarrow{l}(t)=\overrightarrow{a}+t(\overrightarrow{b}-\overrightarrow{a}), t \in ...
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1answer
28 views

Plane that is constructed by vectors

I found the following in my notes: The plane that is constructed by two non-parallel vectors $\overrightarrow{v}$ and $\overrightarrow{w}$ consists of all the points of the form $a ...
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1answer
16 views

Using vector analysis proof that a quadrilateral is in fact a paralelogram

Let D be the midpoint of line segment AC. Let E be simultaneously the midpoint of line segments CB and DF. Please note that the rightmost point is labeled F. Goal: Using vector analysis proof ...
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1answer
48 views

Direction of outward normal differential $\hat n\ ds$

In vector calculus, why is the outward normal differential $\hat n\ ds$ around a closed curve $dy\ \hat i - dx\ \hat j$? Why isn't it $-dy\ \hat i + dx\ \hat j$ or something else?
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1answer
58 views

Line integral over Archimedes's spiral

Compute $\int \limits_{C} F.dr$ for $F(x,y)=(x,y)$ and $C$ is the Archimedes's spiral given in polar coordinates by $r=1+\theta$ for $\theta \in [0,2\pi]$ How do I parametrize this? And how do find I ...
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0answers
16 views

Mappings, Jacobians and inverse mappings in a multivariable calculus textbook

I am currently taking a Calculus III course that introduces the topics below between multivariable differential calculus and multiple integrals. What would be a good reference text for a more ...
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2answers
77 views

What is the gradient of a gradient?

I'm a student, trying to re-derive a result found in a paper by calculating the following in spherical coordinates: $$\mathbf{I}+ \frac{\nabla\nabla}{constant}$$ where $\mathbf{I}$ is a $3x3$ ...
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0answers
12 views

Transform to cylindrical coordinate system

I tried so many approaches , at least give me a hint on how to find The unit vectors $$ \vec{V} = y\vec{i} + x\vec{j} + \frac{x^2}{\sqrt[2]{x^2+y^2})} \vec{k}\ $$
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1answer
36 views

How would you apply the HouseHolder reflections when doing a QR factorization?

I understand the concept of using HouseHolder transformations during QR factorization, but I'm not quite sure how to actually apply them to an example. If we had some matrix, for example $$ ...
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1answer
23 views

Proving a case of Green's Theorem

I'm tasked with showing that $$\iint_D \nabla^2 u dA = \oint_C \frac {\partial u}{\partial n} ds$$ where $\frac {\partial u}{\partial n}$ is the outerward normal derivative. I'm not sure if I can ...
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0answers
12 views

How to parameterize a plane

Given the plane $F: 6x+3y+2z=12$, I shall calculate the flux of a vector $A$field through that plane in the first octant ($x,y,z \geq 0) $. To do this, I can write $F$ as $(x, y, 6-x-\frac32y)$. ...
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0answers
18 views

Estimate tangent plane to function at point using table

I am doing an exam review that has no answer key and am not sure how to approach this question. My approach: If I let $P =$ the vector$ <1, 2, 3.20>, Q = <2, 1.01, 3.11>,$ and $R = ...
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66 views

find the system of linear equations

Find the system of linear equations, that has subspace of solutions similar to linear span of vectors system $a_1, a_2, a_3$, where: $a_1 = (2, 1, -1, 1)$, $a_2 = (1, 2, 1, -1)$, $a_3 = (1, -4, -5, ...
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2answers
38 views

Related Stokes' Theorem

How would you prove the following using Stokes's Theorem? $$ \int_{S}(d\vec{S}\times\vec{\nabla})\times\vec{v}=\oint_{C}d\vec{l}\times\vec{v} $$ I know you pretty much should use a substitution that ...
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0answers
21 views

Closed Line Integral for Scalar Fields Expression Proof Using Stokes' Theorem

I was curious to find an expression that would be Stokes's Theorem, but for scalar fields, i.e. an expression for scalar closed line integrals. Basically I would like to proof the following statement ...
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0answers
28 views

Divergence and curl in curvilinear coordinates

Task: Let $F(R,\theta ,z) = R\widehat R + z\widehat \theta $ in cylindrical coordinates. Calculate div $F$. Let $F(R,\theta ,\varphi ) = {R^2}\widehat R + R\varphi \widehat \varphi $ in spherical ...
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0answers
36 views

Need help identifying PDE with terms of the form $\frac{\partial^2 u1}{\partial x^2 }+\frac{\partial^2 u2}{\partial y^2} \cdots$

So while reading this and some other lecture notes on PDE solving methods, I learnt about Semilinear 2nd order PDEs and briefly got a taste of solving them using the Method of Characteristics ...
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2answers
75 views

Proving vector identities using suffix notation

Show, using suffix notation, the following identity $$\nabla\times\nabla\phi = \mathbf{0}.$$ So we have: \begin{align}\nabla\times\nabla\phi &= \Big[\nabla\times\nabla\phi\Big]_{i}\\&= ...
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1answer
50 views

Change angle of a vector to another vector

Let $\mathbf{x},\mathbf{y},\mathbf{w}$ be the following 3-vectors: $$\mathbf{x}=\begin{pmatrix}x_{1}\\ x_{2}\\ x_{3}\end{pmatrix}\qquad\mathbf{y}=\begin{pmatrix}y_{1}\\ y_{2}\\ ...
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1answer
26 views

Curvilinear Coordinates to Solve Integrals

I get the idea of curvilinear coordinates, but don't understand how it would make corresponding symmetrical integrals easier to solve. How would you evaluate an integral, for simplicity's sake say $$ ...
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0answers
20 views

Gauß's Theorem – Flow Problem

Air flowing with a speed of 0.4 m/s in the direction of a vector $[-1,-1,1]$ goes through a closed loop C joined by the following points in order: $(1,1,0) \rightarrow (1,0,0) \rightarrow ...