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Questions tagged [vector-analysis]

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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163 votes
14 answers
99k views

Why is gradient the direction of steepest ascent?

$$f(x_1,x_2,\dots, x_n):\mathbb{R}^n \to \mathbb{R}$$ The definition of the gradient is $$ \frac{\partial f}{\partial x_1}\hat{e}_1 +\ \cdots +\frac{\partial f}{\partial x_n}\hat{e}_n$$ which is a ...
Jing's user avatar
  • 2,327
9 votes
5 answers
4k views

Divergence of $\vec{F} = \frac{\hat{\mathrm{r}}}{r^{2}}$

Consider the vector field $$\vec{\mathrm{F}} = \frac{\hat{\mathrm{r}}}{r^{2}},$$ then the divergence of this field is: $$\vec{\nabla}\cdot\left(\frac{\hat{\mathrm{r}}}{r^{2}}\right) = 4\pi\delta^{3}(\...
Mohammed Salama Ibrahim's user avatar
26 votes
3 answers
21k views

Anti-curl operator

It is known that if a vector field $\vec{B}$ is divergence-free, and defined on $\mathbb R^3$ then it can be shown as $\vec{B} = \nabla\times\vec{A}$ for some vector field $A$. Is there a way to find ...
Max's user avatar
  • 859
34 votes
6 answers
178k 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 ...
Rcwt's user avatar
  • 832
23 votes
2 answers
56k 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 ...
jinawee's user avatar
  • 2,585
23 votes
3 answers
6k 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 ...
pppqqq's user avatar
  • 2,150
0 votes
2 answers
1k views

Deriving the gradient operator in spherical coordinates

This is a sort of problem where I know what to do but do not completely understand what I am doing. I have been taught how to derive the gradient operator in spherical coordinate using this theorem $$\...
uran42's user avatar
  • 3
82 votes
4 answers
178k views

Gradient of 2-norm squared

Could someone please provide a proof for why the gradient of the squared $2$-norm of $x$ is equal to $2x$? $$\nabla\|x\|_2^2 = 2x$$
user167133's user avatar
3 votes
3 answers
3k views

Gradient of $a^T X b$ with respect to $X$

How can I find the gradient of the term $a^TXb$ where $X$ is a $n \times m$ matrix, and $a$ and $b$ are column vectors. Since the gradient is with respect to a matrix, it should be a matrix. But I ...
Shew's user avatar
  • 1,542
16 votes
2 answers
11k views

$C^1$ function on compact set is Lipschitz

Let $\emptyset\ne A\subset\mathbb{R}^n$ be open and let $f \in C^1 (A, \mathbb{R})$ be a function. Let $\emptyset\ne K\subset A$ be compact and convex. I want to prove that $f$ is Lipschitz on $K$; ...
sequence's user avatar
  • 9,648
60 votes
4 answers
136k 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 ...
Euler....IS_ALIVE's user avatar
46 votes
7 answers
171k 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 ...
Demosthene's user avatar
  • 5,450
11 votes
1 answer
9k views

Does the Divergence Theorem Work on a Surface?

The divergence theorem in $\mathbb{R}^3$ says that the integral of the divergence of a vector field over a solid $\Omega$ in $\mathbb{R}^3$ equals the flux through the surface of $\Omega$ denoted by $\...
user avatar
10 votes
1 answer
2k 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 ...
bolbteppa's user avatar
  • 4,399
10 votes
2 answers
1k views

How can I prove that these definitions of curl are equivalent?

I am reading the book "Div, Grad, Curl, and All that" and I got to the section about curl. In this section, the author defines the curl to be $$ (\nabla \times \mathbf{F})\cdot \mathbf{\hat{...
Robert Lee's user avatar
  • 7,233
25 votes
2 answers
14k views

Why is the gradient always perpendicular to level curves?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a function. A level set is a set of points: $$L(c) = \{x \in \mathbb{R}^n | f(x) = c\}$$ Two vectors $a, b \in \mathbb{R}^n$ are perpendicular, when ...
Martin Thoma's user avatar
  • 9,841
23 votes
2 answers
645 views

Proof of this fairly obscure differentiation trick?

Suppose we're tying to differentiate the function $f(x)=x^x$. Now the textbook method would be to notice that $f(x)=e^{x \log{x}}$ and use the chain rule to find $$f'(x)=(1+\log{x})\ e^{x \log{x}}=(1+\...
user1892304's user avatar
  • 2,818
20 votes
3 answers
13k 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 ...
Sayantan's user avatar
  • 3,418
19 votes
2 answers
49k views

Vector derivation of $x^Tx$

Let $x \in \mathbb{R}^n$ What is $$\frac{\partial}{\partial x} [ x^Tx ]$$ My guess is: $\frac{\partial}{\partial x} [ x^Tx ] = 0$, because $[x^Tx] \in \mathbb{R}^1$, hence a real number as is ...
Mahoni's user avatar
  • 803
18 votes
4 answers
24k views

Is it possible to reverse a gradient ($\vec{\nabla}$) operation?

In calculus, the antiderivative (indefinite integral) can be considered as the reverse operation of a derivative. A gradient yields a vector. Is there a similar way of reversing gradient, as you do ...
syockit's user avatar
  • 772
14 votes
1 answer
2k views

Why is this vector field not conservative, even though it has a potential? (what is the actual definition of a conservative vector field?)

My question is really ''what is the definition of a conservative vector field''? I've consulted 3 textbooks that all say a vector field $\vec{F}$ is conservative by definition if there exists a ...
Rax Adaam's user avatar
  • 1,186
9 votes
2 answers
164k views

Find the Vector Equation of a line perpendicular to the plane.

Question: Find the vector equation $r(t)$ for the line through the point $P = (-1, -5, 2)$ that is perpendicular to the plane $1 x - 5 y + 1 z = 1$. Use $t$ as your variable, $t = 0$ should ...
Riley's user avatar
  • 295
5 votes
3 answers
4k views

Tangent line of a convex function

Let $V : \mathbb{R}^n \rightarrow \mathbb{R}$ be (strictly) convex and continuously differentiable on $\mathbb{R}^n$. Show that for any $y \in \mathbb{R}^n$ and any $x \in \mathbb{R}^n$ $$ V(y) \geq ...
TorqueNoFriction's user avatar
4 votes
2 answers
5k views

Shortest distance between $\vec{V_1}$ and $\vec{V_2}$ with $d=|\frac{(\vec{V_1}\times\vec{V_2})\cdot\vec{P_1P_2}}{|\vec{V_1}\times\vec{V_2}|}|$.

I can find the shortest distance $d$ between two skew lines $\vec{V_1}$ and $\vec{V_2}$ in 3D space with $d=\left|\frac{(\vec{V_1}\times\vec{V_2})\cdot\vec{P_1P_2}}{|\vec{V_1}\times\vec{V_2}|}\right|$...
Dan A's user avatar
  • 49
3 votes
2 answers
6k views

Find plane by normal and instance point + distance between origin and plane

I have a question regarding Vectors; Find the equation of the plane perpendicular to the vector $\vec{n}\space=(2,3,6)$ and which goes through the point $ A(1,5,3)$. (A cartesian and parametric ...
Bak1139's user avatar
  • 3,201
34 votes
8 answers
35k views

What does the symbol nabla indicate?

First up, this question differs from the other ones on this site as I would like to know the isolated meaning of nabla if that makes sense. Meanwhile, other questions might ask what it means in ...
Sebastian Nielsen's user avatar
23 votes
3 answers
16k 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 \...
bolbteppa's user avatar
  • 4,399
19 votes
3 answers
22k 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 ...
user1187139's user avatar
15 votes
1 answer
3k 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 \...
Chris kim's user avatar
  • 850
13 votes
2 answers
11k views

Proof that gradient is orthogonal to level set

When we proved the gradient of a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is orthogonal to the level sets of the function $f(\vec{x}) = c$ for some constant $c$, my professor was quite explicit ...
EuYu's user avatar
  • 41.5k
11 votes
3 answers
6k views

Divergence of matrix-vector product

Suppose that $A$ is a matrix field and that $v$ is a vector field. What is the divergence of the matrix-vector product $A \cdot v$, which is a vector field?
shuhalo's user avatar
  • 7,505
11 votes
2 answers
8k 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 ...
SDiv's user avatar
  • 2,530
10 votes
2 answers
51k views

Del operator in Cylindrical coordinates (problem in partial differentiation)

I am currently reviewing basic vector analysis and trying to understand every single detail, however, I got stuck in some derivation. What I want to show is the following: Given the del operator (...
Meshal's user avatar
  • 779
10 votes
2 answers
6k views

What exactly are pseudovectors and pseudoscalars? And where could I read about them?

I can't find good information on the internet. In my mathematical physics class the definition of a vector was given as: That object with magnitude and direction which doesn't change under ...
ben ari's user avatar
  • 395
6 votes
1 answer
3k views

Why is the determinant of the Jacobian the change of volume factor that comes from changing variables?

I can understand this through various examples found in the internet but I can't quite intuitively understand why the determinant of the derivative(in its most general form)-the Jacobian-gives the ...
TheQuantumMan's user avatar
6 votes
2 answers
6k views

Existence of Partial Derivatives Implies Differentiability [closed]

Doing some self-study. My textbook has this Theorem (see below). I understand it but was hoping for something shorter and more intuitive. Any thoughts?
conums's user avatar
  • 363
5 votes
3 answers
7k views

prove $\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$ using index notation

I'm having some trouble using index notation to prove the identity $$\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$$ The closest I can get is by expanding ...
user78655's user avatar
  • 553
4 votes
1 answer
2k views

Surface integral of normal components summations on a sphere

Whilst studying a book on fluid dynamics I came across a curious footnote comment which is essential in another derivation. The footnote states the following two identities: $$ \frac{1}{|S|}\oint_S\! ...
Robert Manson-Sawko's user avatar
3 votes
2 answers
2k views

Verify the following relationship: $\nabla \cdot (a \times b) = b \cdot \nabla \times a - a \cdot \nabla \times b $

Verify the vector identity: $\nabla \cdot (a \times b) = b \cdot \nabla \times a - a \cdot \nabla \times b $ Given that: $a = (R_a, S_a, T_a)$, $b = (R_b, S_b, T_b)$ and $\nabla = (\frac{\partial} ...
user2250537's user avatar
  • 1,101
1 vote
4 answers
5k views

Showing that $\nabla\times(\nabla\times\vec{A}) = \nabla(\nabla\cdot\vec{A})-\Delta\vec{A}$

I have faced difficulties while trying to prove that $$\nabla\times(\nabla\times\vec{A}) = \nabla(\nabla\cdot\vec{A})-\Delta\vec{A}$$ I don't have any clue how can I start to work with it. Any hint ...
dsadfas's user avatar
  • 67
1 vote
3 answers
2k views

$\nabla U=0 \implies U=\mathrm{constant}$ only if $U$ is defined on a connected set? [duplicate]

How can I prove the following statement? Given a function $U: A \subset \mathbb{R}^n \to \mathbb{R}$ with $A$ connected set, $$\nabla U=\bar{0} \,\,\,\,\,\,\, \forall \bar{x} \in A\implies U=\...
Gianolepo's user avatar
  • 2,507
39 votes
7 answers
5k views

Why isn't the directional derivative generally scaled down to the unit vector?

I'm starting to learn how to intuitively interpret the directional derivative, and I can't understand why you wouldn't scale down your direction vector $\vec{v}$ to be a unit vector. Currently, my ...
rb612's user avatar
  • 3,560
27 votes
2 answers
139k views

Derivative of cross-product of two vectors

In finding the derivative of the cross product of two vectors $\frac{d}{dt}[\vec{u(t)}\times \vec{v(t)}]$, is it possible to find the cross-product of the two vectors first before differentiating?
dtg's user avatar
  • 1,383
18 votes
7 answers
26k views

What is the intuition behind the unit normal vector being the derivative of the unit tangent vector?

I've seen the math, but... It just doesn't make sense to me. How is the slope going to point perpendicular to the vector that is clearly a straight line not going in that direction?
David's user avatar
  • 1,094
16 votes
3 answers
2k views

Existence and uniqueness of Stokes flow

What are the solution existence and uniqueness conditions for Stokes' flow? $$\begin{gathered} \nabla p = \mu \Delta \vec{u} + \vec{f} \\ \nabla \cdot \vec{u} = 0 \end{gathered}$$ Maybe you could ...
Džuris's user avatar
  • 2,590
11 votes
2 answers
4k views

Product rule for scalar-vector product

Let $\mathbf F : \mathbb R^p \to \mathbb R^s$ and $\phi : \mathbb R^p \to \mathbb R$ be differentiable functions. Let the function $\mathbf G$ be defined as follows: $$\mathbf G : \mathbb R^p \to \...
giobrach's user avatar
  • 7,490
8 votes
1 answer
3k views

Every immersion is locally an embedding?

Let $\varphi: M \to N$ be an immersion and let $p$ be a point in $M$. How would I go about showing that there exists a neighborhood $V \subset M$ of $p$ such that the restriction $\varphi|V$ of $\...
user avatar
7 votes
1 answer
1k views

Domains for which the divergence theorem holds

In the book Elliptic partial differential equations of second order written by Gilbarg and Trudinger, I saw the following sentence on page 17 in section 2.4 Green’s Representation: As a prelude to ...
04170706's user avatar
  • 359
6 votes
2 answers
304 views

What is the mechanism of Eigenvector? [closed]

I have studied EigenValues and EigenVectors but still what I can't see is that how EigenVectors become transformed or rotated vectors.
user122358's user avatar
  • 2,712
5 votes
3 answers
4k views

Finding the area between the two ellipses $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$

I'm trying to find the area inside the intersection of the region bounded by the ellipses $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$ using vector calculus methods. I ...
jonathan's user avatar
  • 353

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