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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34
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5answers
2k 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 ...
34
votes
3answers
567 views

Is Stokes' Theorem natural in the sense of category theory?

Stokes' Theorem asserts that for a compactly-supported differential form $\omega$ of degree $n-1$ on a smooth oriented $n$-dimensional manifold $M$ we have the marvellous equation $$\int_M d\omega = ...
31
votes
4answers
2k views

Is there a vector field that is equal to its own curl?

I was wondering if there is a vector field that satisfies the following condition: $$\vec F=\nabla \times \vec F$$
17
votes
3answers
491 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 ...
12
votes
4answers
997 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 ...
11
votes
3answers
2k 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
137 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
3answers
485 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?
10
votes
4answers
24k 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 ...
10
votes
5answers
308 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
287 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
2answers
11k views

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

Suppose that we have two different discreet signal vectors of $N^\text{th}$ dimension, namely $\mathbf{x}[i]$ and $\mathbf{y}[i]$, each one having a total of $M$ set of samples/vectors. ...
9
votes
2answers
3k 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
3answers
428 views

parallelizable manifolds

I know a differentiable manifold $M$ of dimension $n$ is parallelizable if there exist (smooth of course) vector fields $\{X_i\}_{j=1}^n$ which are linearly independent in $T_pM$ at each point $p \in ...
8
votes
3answers
80 views

Why is Green's theorem asymmetric in $x$ and $y$?

Green's theorem is $$\oint_{\partial D} (P\, dx+Q\, dy) = \iint_D dx\,dy \: \left ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)$$ where one can see that the RHS is ...
8
votes
1answer
106 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: $$ ...
8
votes
2answers
172 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 ...
7
votes
6answers
154 views

coordinate free proof that $\text{div}(\nabla f \times \nabla g) = 0$

Let $V$ be a Euclidean $3$-dimensional space. Does there exist a coordinate-free proof that for any two $C^1$-functions $f, g: \mathbb{R}^3 \to \mathbb{R}$ we have $$\text{div}(\nabla f \times \nabla ...
7
votes
1answer
887 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 ...
7
votes
2answers
205 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
280 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
121 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
495 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
6k views

mathematical difference between column vectors and row vectors

I'm writing a mathematical library; and I have an idea where I want to automatically turn column matrices and row matrices to vectors, with all of the mathematical properties of a vector. Answer I'm ...
6
votes
2answers
722 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 ...
6
votes
2answers
8k 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 ...
6
votes
2answers
582 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
5k 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
135 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 ...
6
votes
0answers
62 views

Very difficult surface integral

Compute the surface integral: $$\int_S({x\over \sqrt{x^2+y^2+z^2}}, {y\over \sqrt{ x^2+y^2+z^2}}, {z\over \sqrt{x^2+y^2+z^2}}), \cdot \vec n \ dS$$ where $S: x^3+y^3+z^3=a^3$ The first ...
5
votes
5answers
5k 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
8answers
609 views

Distance of two lines in $\mathbb{R}^3$

Using vector methods show that the distance between two non parallel lines $l_1$ and $l_2$ is given by $$d=\frac{|(\overrightarrow{v}_1 - \overrightarrow{v}_2) \cdot (\overrightarrow{ a}_1 \times ...
5
votes
4answers
936 views

Proof for the curl of a curl of a vector field

For a vector field $\textbf{A}$, the curl of the curl is defined by $$\nabla\times\left(\nabla\times\textbf{A}\right)=\nabla\left(\nabla\cdot\textbf{A}\right)-\nabla^2\textbf{A}$$ where $\nabla$ is ...
5
votes
1answer
54 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
111 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
174 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
313 views

Integral Definition of Exterior Derivative?

Is there a rigorous integral definition of the exterior derivative analogous to the way the gradient, divergence & curl in vector analysis can be defined in integral form? Furthermore can it be ...
5
votes
1answer
197 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
582 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
480 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
2answers
65 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
1answer
95 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
63 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
42 views

Vector Laplace equation with constraint

I want to solve Laplace equation for a vector $\boldsymbol v=(v_x,v_y)$: $$\nabla^2 \boldsymbol{v}=0$$ but under the constraint that $$(1+v_x)^2+v_y^2=1$$ which becomes $v_y = -(2v_x+v_x^2)^{1/2}$. ...
5
votes
0answers
111 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
65 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
125 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 $ ...
5
votes
1answer
268 views

Prove the functions are unique in a volume, vector calculus problem

I am working through the following problem, but finding it hard to know where to go. Using the Divergence theorem and the following identities $\nabla \cdot (A \times B) = B\cdot(\nabla \times A) - ...
4
votes
3answers
487 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
2answers
427 views

If a vector field is divergenceless and curless, is that vector field = 0?

Just a simple question, from the title of the thread, is a vector field = 0 if the divergence is 0 and the curl is 0? I had trouble finding an answer anywhere online, proof of why or why not would be ...