All Questions
Tagged with vector-analysis tensors
144
questions
0
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1
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210
views
Vector Laplacian in Curved Spaces
The vector gradient, $\mathbb{L}$, is defined as
$$
(\mathbb{L} W)^{ij} \equiv \nabla^{i} W^{j} + \nabla^{j} W^{i} - \frac{2}{3} g^{ij} \nabla_{k} W^{k} \,,
$$
where $\nabla_{i}$ is the covariant ...
- 698
0
votes
2
answers
236
views
Power of a second-order tensor
I have the following equation:
$$
\dfrac{\mathbf{V}^T \cdot \mathbf{V}}{\mathbf{V}^T : \mathbf{V}} + \dfrac{\mathbf{V} \cdot \mathbf{V}^T}{\mathbf{V} : \mathbf{V}^T} = \dfrac{\mathbf{D}^2}{\left \| \...
user1012020
1
vote
2
answers
116
views
Why is $\delta\mathbf{u}\cdot \mathrm{div}(\mathbf{\sigma}) = -\mathrm{grad}(\delta\mathbf{u}) \mathbf{:} \mathbf{\sigma}$?
Context is from this deal.ii tutorial. Screenshot of the relevant part is below.
I don't get the transformation from $\delta\mathbf{u}\cdot \mathrm{div}(\mathbf{\sigma})$ to $-\mathrm{grad}(\delta\...
- 866
3
votes
1
answer
6k
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Calculating the Divergence of a Tensor
I am working through a fluid dynamics paper and came across this equation:
$$ \frac{\partial \vec{v}}{\partial t} + \vec{v}\cdot\nabla\vec{v}=\nabla\cdot T - \frac{1}{\rho}\nabla \phi\tag1$$
where T ...
- 1,019
2
votes
1
answer
133
views
What do these tensor partial derivatives mean?
In the Wikipedia page on Ricci calculus the following tensor derivative equation is given:
$$A_{\alpha \beta ..., \gamma}:= \frac \partial {\partial x^\gamma}A_{\alpha \beta ...}.$$
However, what does ...
- 13.4k
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votes
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62
views
Transformations of curvilinear coordinate systems
I'm reading a book covering multiple topics in mathematics and physics. There's a chapter on tensors that had a very curious statement. It says "all curvilinear coordinate systems can relate to ...
- 259
2
votes
1
answer
149
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Extremely complex vector-matrix expression and its differentiation by vector
Given:
$Q=R_z(\psi)R_y(\xi)R_x(\phi)$ - rotation matrix
$\boldsymbol{\theta}=\left[\begin{array}{@{}c@{}} \phi \\ \xi \\ \psi
\end{array} \right]$ - vector of angles
$p=Q\left[\begin{array}{@{}...
- 731
0
votes
3
answers
362
views
Deriving product rule for divergence of a product of scalar and vector function in tensor notation
On page-94 of the 4th edition in the international version of Griffith's Electrodynamic, the following identity is used:
$$ \int \left[ V(\nabla \cdot \vec{E} ) + \vec{E} \cdot \nabla V \right]dV= \...
0
votes
1
answer
379
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$C_{ij}=T_{ijklmn} D_{kl} D_{mn}$ where $T_{ijklmn}$ is a rank 6 isotropic tensor, $C_{ij}$ is symmetric and $D_{ij}$ is antisymmetric
I was doing a question on Tensors and hit a roadblock
The Question:
Suppose that $C_{ij}$ and $D_{ij}$ satisfy the quadratic relationship $C_{ij} = T_{ijklmn} D_{kl}D_{mn}$,
where $T_{ijklmn}$ is an ...
- 23
3
votes
1
answer
148
views
Proving $(\nabla \times \mathbf{v}) \cdot \mathbf{c} = \nabla \cdot (\mathbf{v} \times \mathbf{c})$ using cylindrical coordinates
Assuming the form of divergence in polar coordinates is known, I am attempting to use the following definition of the curl of a vector field to determine the form of the curl in cylindrical ...
- 133
0
votes
1
answer
51
views
How do I reexpress the equation $\nabla \times (\nabla \times gs)) \times (\nabla \times \nabla(f\nabla \cdot t))$?
How do I go about reexpressing
$\nabla \times (\nabla \times bs)) \times (\nabla \times \nabla(c\nabla \cdot t))$
where s and t are vector properties and b and c are scalar.
I don't know where to even ...
2
votes
1
answer
98
views
Expression for Rank 2 Tensor in Vector Notation
How does one write the following expression
$D_{jk} (r_k \delta_{ij} - r_{i}\delta_{jk} - r_j \delta_{ik})$
in matrix notation? Is this just
$\textbf{D} (\textbf{r} \times \textbf{I})$?
- 3,005
0
votes
1
answer
167
views
Gradient in tensor form
I found a problem which had $$\partial_i (A_i \vec{G})= (\vec{\nabla} .\vec{ A} )\vec{G}+ (\vec{A}.\nabla) \vec{G} $$ but my problem is what does $$\partial_i (A_i \vec{B})$$ even mean? it doesn't ...
- 11
3
votes
2
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404
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Help with the gradient in different co-ordinate systems
Let $L(x,y)$ be the linear Taylor series expansion of some function $f(x,y)$. This can be written as $$L(x,y)=f(x_0,y_0)+f_x(x-x_0)+f_y(y-y_0)$$ Or in more compact form as $$L(x,y)=f(x_0,y_0)+\nabla f ...
- 33
0
votes
1
answer
42
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Levi-Civta symbol question
$$\delta_{kl}\epsilon_{ijk}\epsilon_{jki} = \delta_{kl} (\delta_{jl}\delta_{ki} - \delta_{ji}\delta_{kl})$$
$$\delta_{kl}\epsilon_{ijk}\epsilon_{jki} = \delta_{kj}\delta_{ki} -\delta_{ii}\delta_{ji} = ...
- 87