All Questions
Tagged with vector-analysis tensors
144
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259
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Finding the Jacobian of Taylor series of a vector valued function
This question is related to one I asked over on physics stackexchange, but at this point it is purely mathematical in nature and I thought it would make sense to ask here. As the title suggests, I ...
- 123
2
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1
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95
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Solving a linear system in matrix notation
I'm trying to implement the paper "Conformation Constraints for Efficient Viscoelastic Fluid Simulation": https://www.gmrv.es/Publications/2017/BGAO17/SA2017_BGAO.pdf
In implementing eqn. 6, they ...
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3
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1
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150
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Divergence of tensor times vector equals divergence of vector times tensor
Does the following equation hold? $\vec T := \vec T(\vec x)$ is a tensor field, $\vec v := \vec v(\vec x)$ a vector field:
$\text{div} \vec T \cdot \vec a = \text{div}(\vec a \cdot \vec T)$
I think ...
- 193
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If the derivative of tensors are not generally tensors, why does vector calculus work?
There's this chart on Wikipedia
(source: https://en.wikipedia.org/wiki/Matrix_calculus)
Suppose I have the function
$$f(x,y) = x^2y^3$$
and I compute the gradient
$$\nabla f(x,y) = (2xy^3,3x^2y^...
- 4,969
1
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2
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670
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divergence of gradient of scalar function in tensor form
I found simple expression in tensor notation for a divergence of product vector and gradient of scalar function:
$$\operatorname{div}(\mathbf{j}) = 0 \text{, where } \mathbf{j} = \mathbf{m}\times \...
- 29
1
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1
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163
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Tensor chain rule reference request
I am a Maths major student.
Question: Given a function $f:\mathbb{R}^2\to\mathbb{R},$ $g:\mathbb{R}^2\to\mathbb{R}^2$ and $(a_1,a_2)\in \mathbb{N}^2.$
Assume that $f$ and $g$ are infinitely ...
- 15.9k
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300
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Tensor form of $\nabla \times (\phi \vec{V})$
So i'm given to find the tensor/index notation of this Vector identity:
(1) $\nabla\times(\phi{V})=\phi(\nabla\times\vec{V})-\vec{V}\times(\nabla\phi)$
would this just be
$$=\partial_i\epsilon_{ijk}...
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41
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Two questions about vector equations
Q1. How to write the following equation in a form of $\vec{a}= something$?
From the question posted in Physics SE, I found the following process is wrong.
For any $ \vec{v}\ne\vec{0}$
$$ \vec{v}\...
- 45
0
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1
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78
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Inverse for a sum of two matrices
What is the inverse of the 3x3 matrix
$(\vec{a} \vec{a}+cI)^{-1}$
Where $\vec{a} \vec{a} $ is a dyad and $I$ is the identity matrix, $c$ is constant, $\vec{a}=(a_1,a_2,a_3)$ is a 3x1 vector
$\vec{a}...
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What is a neat way to solve $\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{\mathbf{C}}$?
Let $\mathbf{u}:\mathbb{R}^3\to\mathbb{R}^3$ be a smooth enough vector field that satisfies the following equation
$$\nabla\mathbf{u}+\nabla\mathbf{u}^T=\mathbf{C},\tag{1}$$
where $\nabla\mathbf{u}$ ...
- 15k
1
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2
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1k
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Time derivative of scalar function that takes vector as argument
Let's say I have scalar function $\phi$ that is function of some vectors $\vec{\bf{p}}$ and $\vec{\bf{r}}$ such that $\phi = \phi(\vec{\bf{p}}-\vec{\bf{r}})$, also vector $\bf{r}$ is function of time, ...
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8k
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Formula of the gradient of vector dot product
On Wikipedia in the article "Vector calculus identities"
(https://en.m.wikipedia.org/wiki/Vector_calculus_identities)
there are the following two formulas for computing the gradient of vector dot ...
- 61
1
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1
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249
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Proving zero identities in vector calculus with simple arguments involving index counting or symmetry?
Consider the following table describing four second derivative operators.
...
user13618
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Divergence of a dot product of tensor and vector
Hei, I am trying to derive energy equation from Navier-Stokes equation and I come across this:
$$\nabla.(\sigma.v)=(\nabla.\sigma).v +\sigma:\nabla v$$
$\sigma $ is the stress tensor
V :is the ...
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2
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445
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Show that the derivative of a second order tensor gives a third order tensor
Let $U_{i,j}$ be a second order tensor. Show that $\frac{\partial U_{i,j}}{\partial x_{k}}$ is a third order tensor.
I know how to prove that the gradient of a scalar field (which is a tensor of ...
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