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).

Filter by
Sorted by
Tagged with
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
82 votes
4 answers
177k 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
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
60 votes
3 answers
3k 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 = \...
Martin Brandenburg's user avatar
53 votes
6 answers
7k 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 ...
Coffee_Table's user avatar
  • 2,909
49 votes
4 answers
4k 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: $$\vec F=\nabla \times \vec F$$
user avatar
46 votes
7 answers
170k 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
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
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
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
27 votes
2 answers
15k views

Gauss-divergence theorem for volume integral of a gradient field

I need to make sure that the derivation in the book I am using is mathematically correct. The problem is about finding the volume integral of the gradient field. The author directly uses the Gauss-...
Mohamed Ouda's user avatar
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
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
26 votes
1 answer
2k views

Is writing the divergence as a "dot product" a deception?

Suppose we have the following vector field in $\mathbb{R}^3$: $$\vec{F}(x,y,z) = F_x \hat{x}+F_y \hat{y}+F_z \hat{z}$$ where $\hat{x}$, $\hat{y}$, and $\hat{z}$ are unit vectors in each of the ...
5Pack's user avatar
  • 521
25 votes
7 answers
11k views

How would you explain a tensor to a computer scientist?

How would you explain a tensor to a computer scientist? My friend, who studies computer science, recently asked me what a tensor was. I study physics, and I tried my best to explain what a tensor is, ...
closedvolumeintegral's user avatar
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
  • 9,831
24 votes
1 answer
760 views

Tricky surface integral of vector field

The following problem comes from a vector calculus exam. Let $$ S = \left\{ (x,y,z) \in \mathbb{R}: z = e^{1 - (x^2 + y^2)^2}, z > 1 \right\} $$ be an embedded surface with the orientation ...
Sam Skywalker's user avatar
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
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
15k 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
  • 4,389
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
  • 2,808
22 votes
2 answers
476 views

Can a gradient vector field with no equilibria point in every direction?

Suppose that $V:\mathbb{R}^n \to \mathbb{R}$ is a smooth function such that $\nabla V : \mathbb{R}^n \to \mathbb{R}^n$ has no equilibria (i.e. $\forall x \in \mathbb{R}^n : \nabla V (x) \not = 0$). ...
Matthew Kvalheim's user avatar
21 votes
8 answers
4k views

3rd type of vector multiplication beside dot/cross product?

I was reading up on how to find the square root of i , and I learned that multiplication of complex numbers could be viewed geometrically by viewing the complex numbers as coordinates on the complex ...
Hockeyfan19's user avatar
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
20 votes
1 answer
5k views

Determinant of non-square Jacobian

Suppose I have a 3d solid in ${\bf R}^4$ which can be parametrized by the function $F:W\subset{\bf R}^3\rightarrow{\bf R}^4$. Now suppose I want to calculate the volume of this solid. Then naively I ...
Set's user avatar
  • 7,610
19 votes
8 answers
29k views

Can a non-zero vector field have zero divergence and zero curl?

I don't see how. Curl and divergence are essentially "opposites" - essentially two "orthogonal" concepts. The entire field should be able to be broken into a curl component and a divergence component ...
dfg's user avatar
  • 3,901
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
  • 793
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
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
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
17 votes
3 answers
590 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 operator}}\mathbf{V}...
m0nhawk's user avatar
  • 1,779
17 votes
3 answers
6k views

What are examples of parallelizable manifolds, and why does parallelizable correspond to $TM$ being trivial?

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 ...
user avatar
16 votes
2 answers
5k views

Intuitive thinking of differential forms in terms of gradient, divergence and curl?

Is there a way to visualize the differential forms - Like I found that if $\alpha$ is a 1 - form then $d \alpha$ represents curl. and $\sigma$ is a 2 - form then $d \sigma$ represents the ...
BAYMAX's user avatar
  • 4,972
16 votes
4 answers
4k 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 ...
Ahmed Namek's user avatar
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
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
16 votes
2 answers
587 views

When can a vector field be rescaled to have constant divergence?

Suppose $X$ is a smooth vector field on $\mathbb{R}^n$ having divergence $$\nabla \cdot X < 0$$ everywhere. Does there always exist a positive smooth "rescaling" function $g:\Bbb R^n \to (0,\infty)$...
Matthew Kvalheim's user avatar
15 votes
6 answers
26k views

Difference between Gradient Descent method and Steepest Descent

What is the difference between Gradient Descent method and Steepest Descent methods? In this book, they have come under different sections: http://stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf ...
Dozam's user avatar
  • 161
15 votes
6 answers
7k views

Why is gradient in the direction of ascent but not descent?

I understand that differentiation of a function ($\mathbb{R} \rightarrow \mathbb{R} $) at a point is the rate of change in the output for a slight nudge in the input. And this rate of change could be ...
zeal's user avatar
  • 281
15 votes
2 answers
4k views

Why does Green's Theorem require partial derivatives to be continuous

My book (Stewart's Essential Calculus) states Green's Theorem as follows: Let $C$ be a positively oriented, piecewise-smooth, simple closed curve in the plane and let $D$ be the region bounded by $C$...
1729's user avatar
  • 2,157
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
15 votes
0 answers
2k views

Let $f: A \to \mathbb{R^n}$ be of class $C^r$; $Df(x)$ is non-singular for $x\in A$. Show that even if $f$ is not 1-1, the set $B=f(A)$ is open.

Let $A$ be open in $\mathbb{R^n}$; let $f: A \to \mathbb{R^n}$ be of class $C^r$; assume $Df(x)$ is non-singular for $x\in A$. Show that even if $f$ is not one-to-one on $A$, the set $B=f(A)$ is open ...
Julian Lee's user avatar
14 votes
2 answers
9k views

Calculate the Hessian of a Vector Function

I'm working with optimisation. I am trying to obtain the hessian of a vector function: $$ \mathbf{F(X) = 0} \quad \text{or} \quad \begin{cases} f_1(x_1,x_2,\dotsc,x_n) = 0,\\ f_2(x_1,x_2,\...
Abraham Alvarez's user avatar
14 votes
1 answer
15k views

Closed surface integral of the surface's normal vector

Is it true that the surface integral over any closed surface (we are in $\mathbb R^3$) of the normal vector $\hat n$ of that surface, say $\hat n$ is pointing outward, is zero? In other words, is it ...
user3208430's user avatar
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
14 votes
1 answer
9k views

Chain rule for Hessian matrix

Given $f\colon \mathbb{R}^n\rightarrow \mathbb{R}$ smooth and $\phi \in GL(n)$. What is the Hessian matrix $H_{f\circ \phi} = \left(\frac{\partial ^2 (f\circ \phi)}{\partial x_i\partial x_j}\right)_{...
user36047's user avatar
  • 155
14 votes
2 answers
3k 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 ...
Gold's user avatar
  • 26.5k
14 votes
1 answer
2k views

Observation on rot (curl), div and grad on a vector field?

Let $\vec{F}(x, y, z)$ be a vector-valued function describing a vector-field. Then the rotation and divergence of the field are: $\nabla \times \vec{F} = \text{curl}(\vec{F}) = \color{red}{(\frac{\...
Ziezi's user avatar
  • 641
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
13 votes
3 answers
6k 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 is,...
user avatar

1
2 3 4 5
132