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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27
votes
4answers
1k views

The Meaning of the Fundamental Theorem of Calculus

I am currently taking an advanced Calculus class in college, and we are studying generalizations of the FTC. We just started on the version for Line Integrals, and one can see the explicit symmetry ...
16
votes
3answers
472 views

Does taking $\nabla\times$ infinity times from an arbitrary vector exists?

Is it possible to get the value of: \begin{equation} \underbrace{\left[\nabla\times\left[\nabla\times\left[\ldots\nabla\times\right.\right.\right.}_{\infty\text{-times taking curl ...
11
votes
11answers
95k views

How to find perpendicular vector to another vector?

How do I find a vector perpendicular to a vector like this: $$3\mathbf{i}+4\mathbf{j}-2\mathbf{k}?$$ Could anyone explain this to me, please? I have a solution to this when I have ...
11
votes
3answers
1k views

Vector Calculus Identities Using Differential Forms

Is there a nice way to derive $$ \nabla (\vec{F} \cdot \vec{G}) = \vec{F} \times (\nabla \times \vec{G} ) + \vec{G} \times (\nabla \times \vec{F}) + (\vec{F} \cdot \nabla ) \vec{G} + (\vec{G} \cdot ...
11
votes
1answer
120 views

Non-divergence form of a 2nd order PDE

This might be a trivial question but I'm very rusty with regards to calculus and am new to PDEs. How would you write the following second order quasilinear equation in it's non-divergence form: The ...
10
votes
4answers
836 views

Why vector calculus seems inconsistent and vague

I am a senior student of engineering and I have been studying calculus for a while when I reached the part of vector calculus I felt that this part is inconsistent and there is a multiple questions ...
10
votes
5answers
292 views

Why does $\oint\mathbf{E}\cdot{d}\boldsymbol\ell=0$ imply $\nabla\times\mathbf{E}=\mathbf{0}$?

I am looking at Griffith's Electrodynamics textbook and on page 76 he is discussing the curl of electric field in electrostatics. He claims that since $$\oint_C\mathbf{E}\cdot{d}\boldsymbol\ell=0$$ ...
10
votes
1answer
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 ...
9
votes
3answers
438 views

How is “area” a vector?

"We consider Area as a vector." How is an area a vector? Why is that the vector is always normal to the area element?
8
votes
4answers
16k views

Surface Element in Spherical Coordinates

In spherical polars, $$x=r\cos(\phi)\sin(\theta)$$ $$y=r\sin(\phi)\sin(\theta)$$ $$z=r\cos(\theta)$$ I want to work out an integral over the surface of a sphere - ie $r$ constant. I'm able to derive ...
8
votes
2answers
8k views

What is the proof that covariance matrices are always semi-definite?

Suppose that we have two different discreet signal vectors of $N^{th}$ dimension, namely $\textbf{x}[i]$ and $\textbf{y}[i]$, each one having a total of $M$ set of samples/vectors. $\textbf{x}[m] = ...
8
votes
2answers
2k views

Geometric intuition behind gradient, divergence and curl

I learned vector analysis and multivariate calculus about two years ago and right now I need to brush it up once again. So while trying to wrap my head around different terms and concepts in vector ...
8
votes
2answers
136 views

real meaning of divergence and its mathematical intuition

how does divergence which means sink or source equal to ∂Fx/∂x+∂Fy/∂y +∂Fz/∂z.I have been thinking it for a long time and i think "divergence tells us how fast the vector increases when we move apart ...
8
votes
1answer
76 views

Green's identity contradicts Helmholtz theorem

Let F be a vector field on a bounded domain $V \subset \mathbb{R}^3$, which is twice differentiable, let $S := \partial V$. According to the Helmholtz Theorem, F can be decomposed, such that: $$ ...
7
votes
2answers
152 views

How would one arrive at the formulas for divergence and curl?

It has been some years since I've taken multivariable calculus now, but there's something I really never understood: how people would discover the expressions for divergence and curl. I mean, the ...
7
votes
1answer
171 views

The proof of the Helmholtz decomposition theorem through Neumann boundary value problem

As you know, the Helmholtz decomposition theorem is as follows. Let $\Omega$ be open and simply connected, and bounded region in $\mathbb R^3$. Then the smooth vector field $E : \Omega \to ...
7
votes
1answer
115 views

What is the average length of all integral curves of a vector field?

Considering a vector field with a source and a sink in a finite comact space, are there any bounds on the length of the integral curves? Specifically, I am interested in the average length of ...
6
votes
3answers
356 views

What is the name for a vector field that is both divergence-free and curl-free?

Consider a smooth vector field $\mathbf u\colon\Omega\to\mathbb R^3$ defined on an open domain $\Omega\subseteq\mathbb R^3$ such that $\mathbf u$ has zero divergence and zero curl on $\Omega$, that ...
6
votes
2answers
446 views

Special case of the Hodge decomposition theorem

I am trying to prove the following special case of the Hodge decomposition theorem in differential geometry for an $n$ component vector field $V_i$ in $\mathbb{R}^n$. I have very little knowledge of ...
6
votes
2answers
4k views

What is the difference between Green's Theorem and Stokes Theorem?

I don't quite understand the difference between Green's Theorem and Stokes Theorem. I know that Green's Theorem is in $\mathbb{R}^2$ and Stokes Theorem is in $\mathbb{R}^3$ and my lecture notes give ...
6
votes
3answers
111 views

A Weird Contradiction about angular momentum operator in quantum mechanics

I am starting with the standard definition of an angular momentum operator in quantum mechanics given as $$\mathbf{L} = k(\mathbf{r}\times\mathbf{p}) = k(\mathbf{r}\times\nabla),$$ where ...
5
votes
5answers
4k views

What's the geometrical interpretation of the magnitude of gradient generally?

In the following picture, the author of the Field and Wave Electromagnetics shows the geometrical meaning of the direction of the gradient. That is, only by following the direction of the normal ...
5
votes
1answer
617 views

Do the BAC-CAB identity for triple vector product have some intepretation?

As in the title, I was wondering if the formula: $$a\times (b\times c)=b(a\cdot c)-c(a \cdot b)$$ for $\mathbb R ^3$ cross product has some geometrical interpretation. I've recently seen a proof (from ...
5
votes
1answer
52 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 ...
5
votes
4answers
75 views

Proof: Force always perpendicular and motion in a plane implies that the trajectory is a circle

I am looking for a proof for a physics problem. Consider a particle which is subject to a force $\vec{F}(t)$ with $|\vec{F}(t)| = \text{const}$ which is always perpendicular to the velocity ...
5
votes
3answers
170 views

Find the necessary and sufficient conditions on $A$ such that $\|T(\vec{x})\|=|\det A|\cdot\|\vec{x}\|$ for all $\vec{x}$.

Consider the mapping $T:\mathbb{R}^n\mapsto\mathbb{R}^n$ defined by $T(\vec{x})=A\vec{x}$ where $A$ is a $n\times n$ matrix. Find the necessary and sufficient conditions on $A$ such that ...
5
votes
1answer
57 views

Is the helix the unique path with constant curvature and constant torsion?

If a path has both curvature and torsion constant why it is necessarily a helix? How to prove it?
5
votes
2answers
530 views

Proof of the vector calculus identity $\nabla\cdot(u\times v)=v\cdot(\nabla\times u)-u\cdot(\nabla\times v)$

$$\nabla\cdot(u\times v)=v\cdot(\nabla\times u)-u\cdot(\nabla\times v)$$ How can we go about proving the above vector calculus identity component wise?
5
votes
2answers
56 views

line integral: anticlockwise parametrisation in $\mathbb R^3$

Consider $\gamma$ given by the sides of the triangle with vertices $(0,0,1)^t$, $(0,1,0)^t$ and $(1,0,0)^t$. So $\gamma$ runs through the sides of the triangle. Let $f(x,y,z)=(y,xz,x^2)$. I want to ...
5
votes
2answers
514 views

Directional Derivatives using Polar Coordinates

I am having a hard time with this problem on my homework assignment. Here is the problem, and i will show my work below: If $f( x, y) = -2 x^{2} + 3 y^{2}$, find the value of the directional ...
5
votes
1answer
77 views

Geometric proof for triple vector product Jacobi identity

I believe the vector identity $\vec{a} \times (\vec{b} \times \vec{c}) + \vec{b} \times (\vec{c} \times \vec{a}) + \vec{c} \times (\vec{a} \times \vec{b}) = 0$ is called the Jacobi identity and I know ...
5
votes
1answer
83 views

A line integral

If $\mathbf{B}(\mathbf{x})=\rho^{-1}\mathbf{e}_{\phi}$ in cylindrical polars, find: $$\int_{C}\mathbf{B}\cdot\mathrm{d}\mathbf{x}$$ where $C$ is the circle $z=0,\rho=1,\;0\leq \phi\leq ...
5
votes
1answer
60 views

Dot products of three or more vectors

Can't we construct a mapping from $V^3(R^1)$ to $R$ such that $a.b.c = a_{x}b_{x}c_{x}+a_{y}b_{y}c_{y}+a_{z}b_{z}c_{z}$ (a,b,c are vectors in $V^3(R^1)$ ) and more generally $a^n$ , $a.b.c.d.e...$ ...
5
votes
0answers
104 views

How to solve $\mathrm{diag}(x) \; A \; x = \mathbf{1}$ for $x\in\mathbb{R}^n$ with $A\in\mathbb{R}^{n \times n}$?

I would like to solve the following equation for $x\in\mathbb{R}^{n}$ $$\mathrm{diag}(x) \; A \; x = \mathbf{1}, \quad \text{with $A\in\mathbb{R}^{n\times n}$},$$ where $\mathrm{diag}(x)$ is a ...
5
votes
0answers
51 views

Inhomogeneous Wave Equation in 3 dimensions

From section 7.5 in this source, I see that, for $\vec{x} \in \mathbb{R}^3$, if $$ \frac{\partial^2 u\left(\vec{x},t\right)}{\partial t^2} - \nabla^2u\left(\vec{x},t\right) ...
5
votes
0answers
111 views

vector field as integral

Define a vector field $ \vec{f}(\vec{R}) = \oint_C{|\vec{r} - \vec{R}|^2 d\vec{r} }$ where C is a simple closed curve. show that there are constant vectors $ \vec{P} $ and $ \vec{Q} $ such that $ ...
4
votes
3answers
358 views

Shortest length that a vector can have

I came to the following question from a past exam: The vector $v = (k, k, 3 − k)$ depends on a variable $k$. What is the shortest length of the vector $v$ can have? I know that the answer is ...
4
votes
4answers
2k views

Proof of vector calculus identities

Here is the all identities : http://en.wikipedia.org/wiki/Vector_calculus_identities I need help concerning vector functions and indexing notations. Let $\overrightarrow{a}$ be a (smooth) vector ...
4
votes
1answer
155 views

physical meaning of the vector $(A \cdot \nabla) A$

If we consider the vector $\left ( A \cdot \nabla \right) \: B$, we have in Cartesian coordinates $$\left ( A \cdot \nabla \right) \: B = \left ( A \cdot \nabla B_x \right ) e_x + \left ( A \cdot ...
4
votes
1answer
266 views

Leibniz integral rule confusion

Consider a vector field $\mathbf B:\mathbb R\times \mathbb R^3\to\mathbb R^3$. Let there exists some $\mathbf A:\mathbb R\times \mathbb R^3\to\mathbb R^3$ for which $\mathbf B = \nabla \times\mathbf ...
4
votes
1answer
102 views

Multivariable Calculus Integral Proof

This problem is being very difficult for me to solve, I need help. Consider $F:\mathbb{R}^2\rightarrow\mathbb{R}$ of class $C^1$, suppose that the level curves of $F$ are closed and that $\nabla F$ ...
4
votes
2answers
5k views

Proof relation between Levi-Civita symbol and Kronecker deltas in Group Theory

In order to prove the following identity: $$\sum_{k}\epsilon_{ijk}\epsilon_{lmk}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$$ Instead of checking this by brute force, Landau writes thr product of ...
4
votes
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 $\mathbb{R}^n$? $$ \int_{\|x\|=1} e^{2\pi i (x \cdot y)}~dx $$
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 ...
4
votes
1answer
32 views

Show that the sequence $\Omega^0\Bbb{R}^2\ \longrightarrow\ \Omega^1\Bbb{R}^2\ \longrightarrow\ \Omega^2\Bbb{R}^2$ is exact.

I have been posed the following question, which I am unable to answer: Let $a_1,a_2\in\mathcal{C}^{\infty}(\Bbb{R}^2,\Bbb{R})$ be infinitely differentiable functions such that $\frac{\partial ...
4
votes
1answer
95 views

How must I understand concepts equations of physics?

I teach myself mathematics, but those days I wanted to learn about General relativity (not to pursue in it but only to have some background), perhaps because I am very curious to learn why exactly We ...
4
votes
2answers
122 views

Banach space integral via defining it in $X^{**}$ and then proving it's in $X$

Vector-valued integration is something I generally try not to think about very much. I have the impression that it can be a sort of "rabbit hole" of a subtlety if one allows it to be. So, I tend to ...
4
votes
1answer
83 views

Constructing a vector field

I'm doing a list of exercises, but there is one problem I can't solve. Let $$\Omega=\Big\{(x,y,z)\in\mathbb{R}^3 \ | \ \frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}<1\Big\}.$$ Suppose that ...
4
votes
1answer
857 views

vector/tensor covariance and contravariance notation

As I looked over the Wikipedia article on covariance and contravariance of vectors and $\mathbf{v}=v^i\mathbf{e}_i$ is said as a contravariant vector while $\mathbf{v}=v_i\mathbf{e}^i$ is said as ...
4
votes
2answers
5k views

Gradient of a dot product

The wikipedia formula for the gradient of a dot product is given as $$\nabla(a\cdot b) = (a\cdot\nabla)b +(b\cdot \nabla)a + a\times(\nabla\times b)+ b\times (\nabla \times a)$$ However, I also ...