For questions on the vector operators: gradient, curl and divergence.
See
https://en.wikipedia.org/wiki/Gradient
Questions tagged [grad-curl-div]
918 questions
52
votes
7 answers
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 the usual del operator and $\nabla^2$ is the vector…
Demosthene
- 5,548
49
votes
4 answers
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:
$$\vec F=\nabla \times \vec F$$
user204299
45
votes
4 answers
Divergence as transpose of gradient?
In his online lectures on Computational Science, Prof. Gilbert Strang often interprets divergence as the "transpose" of the gradient, for example here (at 32:30), however he does not explain the reason.
How is it that the divergence can be…
onkroy78
- 451
36
votes
4 answers
Why can we treat differential operators as if they behave like algebraic quantities?
In college, I've come across many instances where we multiply a derivative by a function, and the result somehow becomes the derivative of the function i.e $\frac{d}{dx}\times f=\frac{df}{dx}$— as if we're multiplying "operators" with functions in…
pie
- 8,483
28
votes
3 answers
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 $A$ that would satisfy this equation? (I know…
Max
- 919
26
votes
1 answer
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 directions on a Cartesian coordinate system, and $F_x$,…
5Pack
- 531
17
votes
3 answers
Curl in cylindrical coordinates
I'm trying to figure out how to calculate curl ($\nabla \times \vec{V}^{\,}$) when the velocity vector is represented in cylindrical coordinates. The way I thought I would do it is by calculating this determinant:
$$\left|\begin{matrix}
e_r &…
Marcus Loken
- 211
15
votes
2 answers
Using Divergence theorem to calculate flux
Let $W$ be the region bounded by the cylinder $x^2+y^2=4$, the plane $z=x+1$, and the $xy$-plane. Use the Divergence Theorem to compute the flux of $F = \langle z,x,y+z^2 \rangle$ through the boundary of $W$.
So far I've gotten to the point of…
Hendrix
- 1,466
- 1
- 11
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13
votes
2 answers
Find vector field given curl
I have an equation $\nabla \times \vec{B} = \mu_{0}\vec{J}$, where $\vec{J} = \left\langle f(x,y), g(x,y), 0 \right\rangle$ and need to solve for $\vec{B}$.
I've looked elsewhere on here for how to "undo" the curl operator, but every answer I've…
jackarms
- 255
13
votes
0 answers
Is the shear of a vector field related to the exterior differential?
If we have a real vector space $E$ and some inner product $g$, then we can always project any 2nd order tensor onto three subspaces invariant under automorphisms of $E$.
These projections represent its trace, its antisymmetric part and its traceless…
TeicDaun
- 313
11
votes
2 answers
is it necessary that curl of 2d vector is perpendicular to the plane.
I am just confused, help me guys.
The question comes up, because we say that curl is either clockwise or anti-clockwise at a point.
Holy cow
- 1,467
10
votes
1 answer
Numerically compute and clear divergence of discrete vector field
I have a fluid simulation that represents velocity as a vector field in a grid of cells. The cells all have the same width and the same height, but the height is not necessarily equal to the width. I would like to compute the divergence of this…
Gyoo
- 409
10
votes
1 answer
Which operators commute with curl?
Let $X$ be the space of infinitely differentiable maps from $\mathbb{R}^3$ to $\mathbb{R}^3$. Let $C:X\rightarrow X$ denote the curl map. What are all the linear maps from $X$ to $X$ that commute with $C$? For example, rotations and translations…
Mathew
- 1,928
10
votes
2 answers
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{n}} \ \overset{\underset{\mathrm{def}}{}}{=} \lim_{S \to…
Robert Lee
- 7,654
9
votes
2 answers
Potential and Field
So given a field:
$$\vec E(r)=\frac{\alpha(\vec p \cdot \vec e_r)\vec e_r + \beta \vec p}{r^3}$$
where $α, β$ are constants, $\vec e_r$ is the unit vector in the direction $\vec r$, and $\vec p$ is a constant vector.
I'm supposed to find out the…
Tomy
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