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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4
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1answer
51 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 $\mathbb{R}^n$? $$ \int_{\|x\|=1} e^{2\pi i (x \cdot y)}~dx $$
3
votes
2answers
93 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})\\ ...
-1
votes
1answer
43 views

little breakthrough in vector calculus [on hold]

My question is about path independence of work done in a non uniform field.Consider an electric charge at the origin and it will give out non uniform electric fields wrt x and y axis.The present way ...
0
votes
0answers
42 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 ...
0
votes
1answer
25 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 ...
1
vote
2answers
91 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 ...
1
vote
2answers
76 views

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} ...
2
votes
2answers
79 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 ...
0
votes
0answers
12 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 ...
-1
votes
1answer
17 views

Distance between parametric questions and points. [closed]

Find the distance between the line $x=3t-1$, $y = 2-t$, $z=t$, and each of the following points: a) $(0,0,0)$ b $(2,0,-5)$ c) $(2,1,1)$
1
vote
1answer
22 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!
0
votes
0answers
23 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 ...
-1
votes
2answers
32 views

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.
0
votes
1answer
42 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 ...
2
votes
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), ...
0
votes
1answer
47 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 ...
0
votes
1answer
15 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? .
0
votes
1answer
24 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. ...
-1
votes
2answers
35 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 $2:1$.
2
votes
0answers
26 views

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 $v$?
0
votes
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 ...
-1
votes
2answers
45 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} + ...
1
vote
1answer
38 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 ...
0
votes
2answers
24 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 $$F(x,y,z) = (xz - x\cos(z), -yz +y\cos(z), -4 - (x^2 + y^2)) $$ ...
0
votes
2answers
59 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 ...
1
vote
1answer
26 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 ...
1
vote
1answer
57 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 ...
2
votes
1answer
45 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 := ...
1
vote
1answer
29 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 ...
1
vote
2answers
50 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 ...
0
votes
1answer
25 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 ...
0
votes
1answer
32 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 ...
3
votes
2answers
39 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 ...
0
votes
1answer
22 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 ...
0
votes
2answers
34 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 ...
0
votes
1answer
26 views

What is the difference between single and double modulus signs. Do they both mean magnitude?

What's the difference between a set of single modulus and a set of double modulus signs? On textbooks I have seen the magnitude of two vectors vector as |x-y| but I've seen other sites where they've ...
1
vote
2answers
34 views

derivatives of a vector of functions with respect to a vector

Let $\vec W \in \mathbb R^3$. What is the general solution to: $$\frac{\partial}{\partial \vec{W}} \begin{pmatrix} f(\vec W) \\ g(\vec W) \end{pmatrix} $$ I think that in the ...
1
vote
2answers
46 views

Is $\nabla \cdot (\mathbf{a} \mathbf{b}^\mathrm{T}) = \mathbf{b}(\nabla \cdot \mathbf{a})+(\mathbf{a}\cdot \nabla) \mathbf{b}$ correct?

Wikipedia says that the following statement is a vector identity: $$\nabla \cdot (\mathbf{a} \mathbf{b}^\mathrm{T}) = \mathbf{b}(\nabla \cdot \mathbf{a})+(\mathbf{a}\cdot \nabla) \mathbf{b}$$ Where ...
2
votes
1answer
31 views

Levi-Cevita symbols: Why is $\epsilon_{ijk}\epsilon_{pjk}$ equal to $2\delta_{ip}$, but not $0$?

I'm learning vector calculus on my own and sometimes strange things happen that I don't know how I should explain them. We have this famous equality: ...
0
votes
0answers
19 views

Vanishes of second order

I encountered the following statement in an article: Let $f(X) = f(X_1, ..., X_N) = \sum_{|\alpha| \geq 2} {a_{\alpha}X^{\alpha}}$ be a convergent power series which vanishes of second order at ...
0
votes
1answer
24 views

Rewriting integrals over spheres involving $1/|x|$

The following derivation cames from calculations related to the Laplace equation and its fundamental solution. Let $g(x)$ be a test-function (meaning compact support and infinitely differentiable), ...
0
votes
2answers
83 views

In three dimensions, the Laplacian of $1/r$ is $0$ outside the origin

Why does the following hold? $$\Delta_{3}\frac{1}{r}\Bigg\vert_{\mathbb{R}^3 \setminus \left\lbrace 0 \right\rbrace}=0$$
0
votes
0answers
62 views

Gradient is covariant or contravariant?

I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis i , j , k so by invariance nature of vectors, component of gradient ...
1
vote
0answers
40 views

find the angles of a given vector sum

Assume you have n vectors in 2D space, with different fixed magnitudes $l_i$. The problem is to find the angle of each vector such that vector sum is a specific vector. That is, $\sum l_i \cos ...
0
votes
0answers
33 views

Surface Integrals, orientation and parametrizations.

I'm trying to solve the following problem: Integrate $f(x,y,z)=(x,y,z)$ over the surface $z=12$ $x^2 + y^2 \leq 25$ I parametrized the surface with $\sigma (r, \theta) = r \sin(\theta), r ...
2
votes
2answers
48 views

Requirement for continuity of unit normal vector

When considering a subset $\Omega \subset \mathbb{R}^{n}$. If we consider $\nu$, the outward unit surface normal to $\partial \Omega$, what are the requirements of $\partial \Omega$ which will ...
0
votes
2answers
49 views

How can vectors with different units (position, speed, …) “share” the same space?

My question is too stupid to be googled, so I'll ask it here (because I didn't get any answers from google). Context: I have a three dimensional space and the units for $x,y,z$ are given in meters. ...
2
votes
1answer
23 views

Taylor expansion of the electrostatic potential $1/\|\cdot \|$

I have stumbled over this problem several times in electrodynamics, and I just don't get the hang of it. The task is to do a Taylor expansion of $\,f(\vec{x},\,\vec{a}) = ...
0
votes
1answer
32 views

Compute gradient of this expression more quickly

I want to compute $$\vec{n}\cdot \nabla^\prime G(\,\vec{r},\,\vec{r}^\prime)$$ with $$G(\,\vec{r},\,\vec{r}^\prime) = ...
0
votes
0answers
30 views

Just a curious question. What type of mathematics do we need to deal with a vector valued- function that depends on multiple vectors?

We have a scalar valued function that depends on multiple variables (takes in a vector), what about a vector valued function that depends on multiple vectors??