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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On the Definitions of Line and Surface Integrals

When motivating the definitions of line and surface integrals, one usually defines the length and area elements \begin{align*} ds &:= \| \vec{r}^{\, '}(t) \| dt, \\ dA &:= \| \vec{\Sigma}_{u} ...
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37 views

Vectors - collinear and perpendicular

A bird is at point P whose coordinates are (4, -1, 5)m. The bird observes two points $P_1$ and $P_2$ having coordinates (-1,2,0) and (1,1,4) respectively. At time t = 0, it starts flying in the plane ...
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62 views

Calculate integral involving product of curls

I want to show that $ \frac{1}{k^2} \int \, (\nabla \times \vec M)\cdot (\nabla \times \vec M)^* d^3 r= \int \vec M \cdot \vec M^*d^3 r$. $\vec M$ is defined as $\vec M_{jm} = i\frac{m}{\sin ...
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18 views

Diagonalization of matrix using change of variables

In linear algebra, we know that a system of equations $AX=b$ can be easily solved if $A$ is found to be of diagonal nature. If however $A$ is not diagonal but can be changed into a diagonal form by ...
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269 views

Proof of trigonometric identity using vector calculus

Question: Using vector calculus, show that $\sin (A+B) = \sin A \cos B + \cos A \sin B$ I have no idea how to even attempt the question. A small hint to help me get started would be greatly ...
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1answer
40 views

Integration against divergence free vector fields

Let $\chi:\Omega\to \mathbb{R}^n$ be a vector field on a bounded, smooth domain $\Omega \subset \mathbb{R}^n$. Assume that for any divergence free vector field $\eta:\Omega \to \mathbb{R}^n$ we have ...
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24 views

finding volume of the cone by using the dot product

Vector u = i +j -2 k and v = 2 i + 3 j can be used to form a cone. The cone forms by rotating u about v through 360 degrees. Find the volume of the cone. I drew the diagram already, I need the r and ...
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43 views

Gradient of real part = real part of gradient?

Suppose f(x,y,z) maps $\mathbb{R}^3\rightarrow\mathbb{C}^1$. That is, it takes in three real numbers and spits out a complex number. Does the following always hold: $$\vec\nabla ...
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1answer
23 views

Rotating vector functions

Suppose you have a vector function of space: $\vec{E}(x,y,z)=Ex(x,y,z)\hat{x}+Ey(x,y,z)\hat{y}+Ez(x,y,z)\hat{z}$. Suppose now you want to rotate the whole vector function by using a unitary rotation ...
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31 views

dot product with unknowns of the vectors

$a = (8 , y)$ $b = (2, 3)$ $c = (x, y)$ if $a \cdot c = 10$ and $b \cdot c = 8$, find the values of x and y. I did up to $10 = 8x i + y^2 j$ $8 = 2x i + 3y j$
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251 views

Intuition behind curl identity

Is there any clear intuition behind the identity $$ \nabla\times (\nabla\times A)=\nabla (\nabla \cdot A)-\nabla ^2A $$ Though the result is useful and not difficult to derive, it doesn't quite ...
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1answer
26 views

Calculating the divergence of a central vector field.

I am trying to calculate the divergence of a central electric field, namely the electric field due to a point charge and my book begins like this: http://imgur.com/bW9tPEZ However in the last line I ...
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1answer
53 views

What am I doing wrong in this volume integral (divergence theorem)?

I'm learning about the divergence theorem. If I have a vector function $f(x,y,z)=\sqrt {x^2+y^2} \cdot (x,y,z)$ and I want to get $\iint\limits_A f(x,y,z) \, d A $ (easy to evaluate, but I thought I'd ...
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0answers
27 views

Laplacian in arbitrary spherical coordinates?

assume that $f: \mathbb{R}^n \rightarrow \mathbb{R}$ is a function that just depends on the radius vector $r$, so no angular dependence(!). Can we say what $\Delta f$ is in arbitrary coordinates $r ...
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2answers
87 views

Definite integral-dot product

I have an integral equation containing dot product $$\int_{0}^{L} \left(\frac{a}{L}.b(s)\right)\mathrm ds\tag 1$$ Data Given a is a constant vector of size 3 b(s) is a varying vector of size 3 " . ...
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37 views

Farthest vector direction relative to other vectors

I wished to know the cheapest computational means (be it analytical or numerical) to find the vector from origin (normalised or not; I do not care about its magnitude) given any arbitrary set of ...
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2answers
50 views

What is the simplest way to understand $\nabla \cdot (\nabla \times \textbf{A} ) =0$?

Of course one can prove it by brute force. But how to Understand it, in the simplest way? Another issue is $\nabla \times (\nabla f) =0 $.
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1answer
57 views

Spherical integral

Let $y \in \mathbb{R}^n$ be fixed. Is there a nice expression for the following integral taken over the unit sphere in $\mathbb{R}^n$? $$ \int_{\|x\|=1} e^{2\pi i (x \cdot y)}~dx $$
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117 views

$\Delta \vec{v}=0$ implies $\nabla\cdot \vec{v}=\nabla\times \vec{v}=0$?

\begin{align} \Delta\overrightarrow{v}&=\nabla(\nabla\cdot\overrightarrow{v})-\nabla\times(\nabla\times\overrightarrow{v})\\ ...
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0answers
50 views

Plotting parametric form of a gradient

This is driving me batty. I'm trying to figure out how to plot the gradient of a circle function (is that a vector field?) in parametric form. I don't understand what values to plug in to a get a ...
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1answer
27 views

Why is the derivative Df(p) defined to $ \in \Lambda^1 (\mathbb{R}^n) $, or how is it a 1-form?

I know that obviously the differential operator D would be a differential form through the word differential. But in Spivak Calculus on Manifolds he defines a k-form w $ \in \Lambda^k ...
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2answers
98 views

Using Stokes theorem to integrate $\vec{F}=5y \vec{\imath} −5x \vec{\jmath} +4(y−x) \vec{k}$ over a circle

Find $\oint_C \vec{F} \cdot d \vec{r}$ where $C$ is a circle of radius $2$ in the plane $x+y+z=3$, centered at $(2,4,−3)$ and oriented clockwise when viewed from the origin, if $\vec{F}=5y ...
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derivative formula $\nabla \times (\mathbf{a} \times \mathbf{r}) = \nabla \cdot(\mathbf{a} \wedge\mathbf{r}) = (n-1)\mathbf{a}$

Assume $\mathbf{r}=\mathbf{x}−\mathbf{x}′$ is the position vector in $\mathbb{R}^n$, for constant $\mathbf{a}$, we have $$\nabla \times (\mathbf{a} \times \mathbf{r}) = \nabla \cdot(\mathbf{a} ...
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2answers
94 views

derivative $\nabla \frac{\mathbf{r}}{r^k}$ in the context of Geometric Calculus

Suppose $\mathbf{r} = \mathbf{x - x'}$ is the position vector in $\mathbf{R^n}$, and $r = |\mathbf{r}| = |\mathbf{x - x'}|$. Do we have $\nabla \frac{\mathbf{r}}{r^k} = \frac{n-k-1}{r^k}$ or $\nabla ...
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0answers
14 views

Method of Characteristics (Change of Co-ordinates)

Here below is the notes about the change of co-ordinates from $xy$-plane to $\xi\eta$-plane. I wanna ask for why dot product works for the change, i.e. $\xi=(x,y) \cdot (a,b)$ and $\eta=(x,y) \cdot ...
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1answer
24 views

How to calculate length and area for this curve?

$C : x^{2/3} + y^{2/3} = 1$ I'm stuck, so any tip will be helpful Thanks in advance!
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25 views

How to calculate the flow of fluid through this closed surface?

Considering a fluid whose velocity field is $\vec{v}(x,y,z)= (y^{3}e^{-z^{2}} + x)\vec{i} + (ze^{x} + y^{2})\vec{j} + (cos(x^{2}+ y^{2}) +2z)\vec{k}$ Calculate the flow of fluid through the closed ...
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geometry proof with triangles using vector

in a triangle ABC, P, Q are points on AB and R, S are points on BC such that AP=PQ=QB and CR=RS=SB. Show that PR bisects AS.
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48 views

How to calculate this area? (portion of a sphere inside a cylinder )

The area of ​​the portion of the sphere $ x^{2} + y^{2} +z^{2} = 1$ located inside of the cylinder $x = x^{2} + y^{2}$, and above the plane $z = 0$. I'm stuck, so any tip will be helpful Thanks in ...
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1answer
16 views

On the Continuity of the Jacobian of a diffeomorphism

Let $\phi:U\longrightarrow V$ be a diffeomorphism between the open sets $U, V\subseteq \mathbb R^n$. Provided $J\phi(x)\neq 0$ for all $x\in U$ we have a map $$J\phi:U\longrightarrow GL_n(\mathbb R), ...
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1answer
48 views

Flow of fluid through a really tricky closed surface S (divergence theorem)

Considering a fluid whose velocity field is $\vec{v}(x,y,z)= (y^{3}e^{-z^{2}} + x)\vec{i} + (ze^{x} + y^{2})\vec{j} + (cos(x^{2}+ y^{2}) +2z)\vec{k}$ Calculate the flow of fluid through the closed ...
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1answer
18 views

Does someone knows how calculate scalar product of two gradients and put the result in terms of nabla operator? .

$(\vec\nabla{\gamma})\cdot(\vec\nabla{\gamma})$ = ? Does someone knows how calculate scalar product of two gradients and put the result in terms of nabla operator using indicial notation? .
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1answer
27 views

Metric for vector sets

I am currently working on a classification algorithm. Each class is represented by a set of 3D vectors. The cardinality differs for each class. The order of the vectors in a set is completly random. ...
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41 views

proof of parallelogram using vector and midpoint

$OPQR$ is a parallelogram. $T$ is the midpoint of $OR$. Show that $QT$ cuts the diagonal $PR$ in the ratio $2:1$.
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Existence of gradient perpendicular to a vector field

Let $v$ be a divergence-less vector field (in $\mathbb{R}^3$). When can we find a non-constant scalar function $f$ so that $\nabla f$ is perpendicular to $v$?
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1answer
35 views

Finding a closed line integral using Stokes' Theorem

Find the line integral $\int_C \vec{F} \cdot \vec{dS}$, where $C$ is the circle of radius 3 in the $xz$-plane oriented counter-clockwise when looking from the points $(0, 1, 0)$ into the plane and ...
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53 views

Evaluate the flux integral [closed]

Evaluate the flux integral $$ \int\!\!\int_{S} {\rm curl\left(\vec{F}\right)} \cdot \vec{dS} $$ where $$ \vec{\rm F}(x, y, z) =\langle xe^{y^2}z^3 + 2xyze^{x^2 + z}, x + z^2e^{x^2 + z}, ye^{x^2+z} + ...
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1answer
40 views

Determine flux using Gauss's theorem

Here's the task I'm a bit confused with: Find using Gauss's theorem the outward flux seen from the point $\left(\begin{matrix}0\\0\\1\end{matrix}\right)$ of the vector field ...
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2answers
25 views

Finding a value R that maximizes the flux a vector field over half a sphere of radius R

Sorry for the bad title, couldn't think of a less convoluted way of writing it. I have to find $ R\in \mathbb{R}$ so that the flux of $$F(x,y,z) = (xz - x\cos(z), -yz +y\cos(z), -4 - (x^2 + y^2)) $$ ...
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62 views

Using Stokes theorem to find the integral of a vector field over the curve of intersection of two surfaces

Find $\int_C{ \vec{F} \cdot \vec{dr}},$ where $F(x, y, z) = \langle 2 x^2 y , 2 x^3 /3, 2xy\rangle$ and $C$ is the curve of intersection of the hyperbolic paraboloid $z = y^2 - x^2$ and the ...
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1answer
29 views

Proving that a field satisfies stokes theorem.

The field is the classic $$F (x,y,z) = \left( \frac{-y}{x²+y²}, \frac{x}{x²+y²},0\right)$$ And the surface is the space between $x² + y² =1$ and $x+y=1$ at $z=0$ Since $ \nabla \times F = 0 $ and ...
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1answer
62 views

If $M$ is a compact manifold what does $\partial M$ mean?

In the generalized form of stokes theorem it states that the integral of the $k+1$ differential form of an operator over a compact manifold $M$ is equivalent to the integral of the $k$ differential ...
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1answer
53 views

Differentiation of scalar fields using tensor notation

I'm learning tensor calculus to understand differential geometry. Please verify if I've understood how to employ Einstein's sum convention and index notation correctly. Suppose that $\varphi := ...
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1answer
32 views

Vectors with given angle and magnitude

Give an example of vectors $\mathbf{v}$ and $\mathbf{w}$ such that the angle between $\mathbf{v}$ and $\mathbf{w}$ is $\frac{2\pi}{3}$ and $\|\mathbf{v} \text{ x } \mathbf{w}\|=\sqrt{3}$. Should I ...
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2answers
52 views

Verification of the Stokes theorem for the surface that is a part of a cone

Let $S$ consist of the part of the cone $z=(x^2+y^2)^{1/2}$ for $x^2+y^2\leq9$ and suppose $${\bf A}=(-y,x,-xyz).$$ Verify that Stokes theorem is satisfied for this choice of $\bf A$ and $S$. In ...
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1answer
29 views

Matrices as sets of vectors

What exactly does it mean when someone says a matrix may be intrepreted as a set of vectors? As in: "A matrix can be considered a set of vectors, organised as rows or columns" It seems it would only ...
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1answer
47 views

Finding the area of a triangle from vertices? Linear Algebra

I pretty much did this problem, but I failed to get the few last blanks where they ask the area. Its confusing, they say its half the volume of matrix (u v w) in the start of the question. which means ...
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2answers
51 views

how to prove gradients vectors are the same in polar and cartesian coordinates.

Suppose $T=T(r,\theta)=G(x,y)$ How do you prove $\nabla T(r,\theta)=\nabla G(x,y)$? I can think of some arguments in favor of this equality, but I want an actual proof or a very good intuitive ...
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1answer
34 views

Areas of tetrahedron surfaces - how to calculate?

Reading up on Cauchy's stress theorem, I have stumbled over the so-called Cauchy tetrahedron, which is an important part of the theorem's proof. The following is cited straight from Wikipedia, but a ...
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2answers
35 views

Show that there is no vector field with curl $x \hat i + y \hat j + z \hat k$.

I have no idea how to prove this. By assuming the field has the curl I get these 3 equations: $$x = \frac{\partial F_{3}}{\partial y} - \frac{\partial F_{2}}{\partial z}$$ $$y = \frac{\partial ...