All Questions
Tagged with curl or grad-curl-div
936 questions
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Strain Tensor Based on Clifford Algebra
Let $u=u_i e_i$ be the displacement field of a continuum body. Then the displacement gradient tensor H based on classical formulation is given by $H=\nabla u = u_{i,j}\, e_i \otimes e_j$, where $\...
2
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1
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116
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What are the complex fixed points of the curl operator? ($\nabla \times \vec v = \vec v$)
Let $\vec{v}=(F_x,F_y,F_z)$, where the components are functions $\mathbb C^3 \to\mathbb C$ and the subscript simply denotes the coordinate. I am curious in finding non-trivial complex $\vec{v}$ such ...
- 674
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Continuity of surface curl
Question
Let $\Omega\in\mathbb{R}^3$ and $\Gamma=\partial\Omega$ its boundary. For any point $\mathbf{x}\in\Gamma$, let $\mathbf{n}_\mathbf{x}$ be the outward unit normal to $\Gamma$ at the point $\...
- 1,108
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101
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Can you express divergence free vector fields as a cross product of gradients?
I know that a divergence free vector field in $R^3$: $A$ as a curl: $\nabla\times B$.
My question is: if we have a vector field $A$ in $R^3$ such that $\nabla\cdot A =0$, does there exist scalar ...
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128
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Is there any sign mistake in computing the divergence in general coordinates?
$\newcommand{\pd}{\partial}$
$\newcommand{\wdw}{\wedge \cdots \wedge}$
$\newcommand{\cdc}{, \cdots ,}$
I'm trying to compute the divergence in general coordinates for the case of pseudo Riemannian ...
- 4,722
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44
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Divergence of product of measure and vector field
We say that a scalar field $v : \mathbb{R}^d \to \mathbb{R}$ is a weak divergence of the vector field $u : \mathbb{R}^d \to \mathbb{R}^d$ if for all test function $\phi$, one has:
\begin{equation}
...
- 618
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2
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134
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Is there terminology for the "line integral" in the normal/divergence form of Green's Theorem?
For a simply-connected planar region $R$ whose boundary is a simple, piecewise-smooth curve $C = \partial R$ with parameterization $\mathbf{r}$ and tangent vector $\mathbf{T}$, and a two-dimensional ...
- 19.6k
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Upper bound of the sum of errors for minimizing an L-smooth convex function using gradient descent method
$f(x)$ is an L-smooth convex function, $x_*$ is the minimizer of $f(x)$.
We define $x_{k+1}=x_k-\frac{1}{L}\nabla f(x_k)$.
We can show that $$f(x_0)-f(x_*)\le\frac{L}{2}|x_0-x_*|^2$$ and $$\sum_{k=1}^{...
- 3
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23
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Computing the curl of a vector function defined as an integral: discrepancy in vorticity for a point torque in a wedge compared to planar wall limit
I am trying to solve the Stokes equations
$$
-\boldsymbol{\nabla}p + \eta \boldsymbol{\nabla}^2 \mathbf{v} +
\tfrac{1}{2} \, \mathbf{T} \times \boldsymbol{\nabla} \, \delta(\mathbf{x}-\mathbf{...
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Intuition behind Stokes' Theorem surface independence
Divergence theorem is intuitive to me in that the sum of all of the sources and sinks inside of a volume must equal the net "flow" through the boundaries of said volume: if there is more &...
- 11
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21
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Function to represent a direction dependent density or opacity
Suppose a point $\vec{x}\in\mathbb{R}^3$. I would like to know how to formulate the following, which I try to explain with a light beam going through the point.
The light beam is dimmed or, similarly,...
- 675
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47
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Computing the derivative of a quadratic form using the chain rule
Consider the quadratic form $f(x) = \langle x, Ax\rangle$ for $x \in \mathbb{R}^n$. I know that
$$\nabla f = (A+A^T)x$$
but I want to compute this using the chain rule. The answer is already given ...
- 969
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2
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95
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Problem with vector calculus - gradient in spherical coordinates
While deriving the electric field from a dipole source, from the notes I am following I am required to process the following vector operation:
$$
\nabla \left(\frac{e^{jkr}}{r}\mathbf n\cdot \mathbf p\...
- 3
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61
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Laplacian of vector spherical harmonic
According to this wiki page here, given the vector spherical harmonic
$$
\boldsymbol{\Phi}_l^m = \mathbf{r} \times \nabla Y_l^m
$$
its Laplacian is,
$$
\nabla^2 \boldsymbol{\Phi}_l^m = -\frac{l(l+1)}{...
- 1,234
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45
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Divergence of directional derivative of a vector
Let $(a \cdot \nabla) b$ be the directional derivative of the vector field $b$ in the direction of $a$, and let
$$
(J_a)_{ij} = \partial_j a^i
$$
be the Jacobian matrix of $a$. Then I am interested in ...
- 1,234