All Questions
Tagged with vector-analysis tensors
144
questions
0
votes
0
answers
34
views
Changing coordinates of $2$nd order partial operators
Let's work in $\mathbf R^n$. If we want to change coordinates $\mathbf x\to\mathbf r$, with them related like
$$
\mathbf x=\mathbf x(\mathbf r)
$$
Then, the second order generic partial operator in ...
- 2,638
0
votes
0
answers
115
views
tensor identity
The tensor $t$ is defined in terms of a scalar $\varrho$ and the two vectors $v$ and $u$ (and derivatives) as follows:
$$t_{i j}:=\varrho v_i u_j-\frac{1}{2} \varrho \varepsilon_{i lm} v_l (\...
0
votes
1
answer
57
views
$\nabla\cdot(f\vec{g})=f\nabla\vec{g}+\vec{g}\cdot\nabla f$ using Levi-Civita
I need to prove the following equality: $\nabla\cdot(f\vec{g})=f\nabla\vec{g}+\vec{g}\cdot\nabla f$.
I know that proving this equality using the properties of $\nabla$ and $(\cdot)$ is easy, what the ...
- 135
2
votes
1
answer
75
views
geometric interpretation on covariant derivateve in curvilinear coordinates.
I'm having trouble understanding what does the covariant derivative do in a coordinate system where we have changing basis vectors. I always thought it was giving us the change in coordinates while ...
- 379
25
votes
7
answers
11k
views
How would you explain a tensor to a computer scientist?
How would you explain a tensor to a computer scientist? My friend, who studies computer science, recently asked me what a tensor was. I study physics, and I tried my best to explain what a tensor is, ...
2
votes
1
answer
67
views
Differential surface element and nabla operator
If we have the vector field $\vec{u}=\vec{A}\times \vec{v}$, where $\vec{A}=\text{const.}$ and we integrate over some closed curve, by using Stokes' theorem we get:
$$
\begin{align}
\oint_{\partial S}...
- 379
0
votes
1
answer
44
views
If $S$ is a symmetric matrix, then rewriting $\int{S:\nabla \phi} dx$
I am trying to prove that:
If $S$ is a symmetric matrix, then one can rewrite $\int{S:\nabla \phi} \text{dx}$ as $\int{S:D(\phi)} \text{dx}$, where $D(\phi)$ is the symmetric gradient of $\phi$.
Any ...
- 194
1
vote
1
answer
104
views
Exercise 5.22 in the book Geometrical Methods of Mathematical Physics (by Bernard Schutz)
I can't figure it out about the Exercise 5.22 in the book Geometrical Methods of Mathematical Physics (by Bernard Schutz).
Could any one give me a help? Thanks.
($\bar{V}$ means vector V. )
Exercise ...
- 63
0
votes
2
answers
119
views
What is the gradient in this skewed coordinate system?
Consider two bases $e_i$ and $f_i$ of $\mathbb{R}^2$ defined by:
$$\begin{aligned}
(f_1,f_2) &= (e_1,e_2)\cdot F\\
\begin{pmatrix}1&-1\\1&1\end{pmatrix} &= \begin{pmatrix}1&...
- 98
0
votes
0
answers
52
views
Intuition about the divergence of a vector field in non-orthogonal basis
My textbook defines the divergence of a vector field in a non orthogonal constant basis the following way:
$$div(\vec{u})=\vec{a}^i\cdot\frac{\partial \vec{a}_ku^k}{\partial x^i}=\frac{\partial u^i}{\...
- 379
1
vote
0
answers
45
views
How to integrate by parts the vectorial product between a vector and a gradient [closed]
I am having a problem trying to check if the identity below should be positive or negative.
$$ \int\; \boldsymbol{A} \times (\boldsymbol{\nabla} a ) = \pm \int (\boldsymbol{\nabla} \times \boldsymbol{...
- 11
0
votes
0
answers
51
views
Differential operators of tensor fields. Hamilton operator
The very first thing my textbook says is that the Hamilton operator is defined as:
$$\vec{\nabla}=\vec{a}^i\nabla_i$$
Where $\nabla_i$ is the covariant derivative and " $\vec{a}^i$ is the ...
- 379
2
votes
2
answers
93
views
Derivation or intuition on the covariant derivative for higher rank tensors
So the derivation in my textbook for the covariant derivative of a vector field $\vec{u}$ in curvilinear coordinates $\xi^k$ is the following:
$$\frac{\partial \vec{u}}{\partial\xi^j}=\frac{\partial (...
- 379
0
votes
0
answers
25
views
Why does the invariant 1-tensor integral gives 0 for any volume?
I was going through David Tong's vectors calculus notes. In Chapter 6.1.3 (Invariant Integrals) he gives the example
Here are some examples. First, suppose that we have a 3d integral over
the ...
- 1
0
votes
0
answers
92
views
Averaging the components of a unit vector over a circle
I have a following problem:
"Calculate the average values of the products of the components of the unit vectors: $$\langle n_i \rangle, \langle n_i n_j \rangle, \langle n_i n_j n_k \rangle, \...
- 1
0
votes
1
answer
59
views
Metric Tensor Grid
Let, we are in a 2d metric where $g_{xx}=1, g_{yy}=x^2$, therefore $|e_x|=1$, $|e_y|=x$. If we try to draw the metric in a grid - it looks something like the image I uploaded. Note that, along the X ...
- 109
1
vote
1
answer
140
views
Covariant and partial derivative of a vector field (not component)
Is the covariant derivative of a vector field (not the components of a vector) same as the partial derivative?
I am adding a screenshot from page 69 from General Relativity: An introduction for ...
- 109
0
votes
0
answers
61
views
Confusion between covariant and partial derivatives
Let, we are in 1d cartesian space with metric $g_{xx} = x^2$. Let we have a vector $v = 1/x e_x$. Since the vector is designed to shrink its components as the basis grows - its total length will ...
- 109
0
votes
2
answers
101
views
The difference between two indices suffix notation
Recently reading a set of lecture notes on vector calculus, which is a topic I am already familiar with. However during this I came across this representation of the gradient vector...
$$\frac{\...
- 80
7
votes
2
answers
289
views
What is $\left ( \vec{\nabla} \times \vec{A} \right ) \cdot \left ( \vec{\nabla} \times \vec{A} \right )$?
I'm trying to rewrite $\left ( \vec{\nabla} \times \vec{A} \right ) \cdot \left ( \vec{\nabla} \times \vec{A} \right )$ in some other way. I tried using Levi-Civita symbol and Kronecker delta, but I'm ...
- 120
5
votes
3
answers
259
views
What are the linear transformations that preserves the cross product, i.e. $ R(v\times w) = (Rv) \times (Rw), \forall v,w \in \mathbb{R}^3 $
Let us just focus on $\mathbb{R}^3$ currently. We study the set of all $3\times 3$ matrices $R$ satisfying
$$
R(v\times w) = (Rv) \times (Rw), \forall v,w \in \mathbb{R}^3
$$
where $\times$ is the ...
- 658
1
vote
0
answers
40
views
Whats the significance of $g^{-1}$ (the inverse metric) appearing in tangential projection?
Let $M \subseteq (\mathbb{R}^n,g_E)$ be an embedded submanifold, with the embedding $F : M \to \mathbb{R}^n$. It is well known (c.f. Lee, doCarmo) that the covariant derivative on $M$ with respect to ...
- 198
4
votes
2
answers
413
views
Proving $A\times(\nabla\times B)+B\times(\nabla\times A)=\nabla(A\cdot B)-(A\cdot\nabla)B -(B\cdot\nabla)A$ with Einstein summation
So, I'm seeking to prove the below identity, for $A,B$ vectors fields in $\mathbb{R}^3$:
$$A \times (\nabla \times B) + B \times (\nabla \times A) = \nabla (A \cdot B) - (A \cdot \nabla)B - (B \cdot \...
- 45.9k
0
votes
1
answer
241
views
Divergence of a Tensor Field
Given a tensor field $\hat{\tau}$, I wish to calculate $\nabla\cdot\hat{\tau}$.
My first question: is this actually the divergence of the tensor field? Wikipedia seems to differentiate between div$(\...
- 1,019
0
votes
0
answers
126
views
Tensor and Gauss divergence theorem
I am trying to see whether, in spherical coordinates,
$$\int \left( \boldsymbol{r} \times \boldsymbol{\nabla} \cdot \boldsymbol{T} \right) \cdot \boldsymbol{e}_z dV$$ where $T$ is a 2D symmetric ...
- 21
0
votes
0
answers
120
views
How do I simplify $\delta_{ij} \delta^{jk}$?
How do I simplify $\delta_{ij} \delta^{jk}$?
I know that $\delta_{ij} \delta_{jk}=\delta_{ik}$, but what do I do if the there's a Kronecker Delta symbol with upper indices and one with lower indices?
- 93
0
votes
0
answers
34
views
Can you define a tensor by integrating one vector with respect to another?
I was reading this question, simply I was wondering about integrating a vector with respect to another vector field. In the question, the OP asks if the following quantity has any sensible meaning:
$$\...
- 211
6
votes
0
answers
207
views
'Tensor Calculus' by J.L. Synge and A. Schild (1979 Dover publication) . Exercise 12 page 110.
I'm solving all the exercises of 'Tensor Calculus' by J.L. Synge and A. Schild (1979 Dover publication) . Till now everything went smooth, but now I'm stuck at exercise 12. page 110 of the third ...
- 116
0
votes
1
answer
143
views
Tensor calculus - product of metric tensor and second covariant derivative of a scalar (Laplace-Beltrami operator)
I am trying to prove the following.
Suppose we have a scalar function $\phi$ (sufficiently differentiable), the metric tensor $g_{ij} = \dfrac{\partial y^\alpha}{\partial x^i}\dfrac{\partial y^\alpha}{...
0
votes
1
answer
444
views
Tensor calculus - gradient of the Jacobian determinant
Given an invertible coordinate transform between a set of coordinates $\{y^1, ..., y^n \}$ and $\{x^1, ..., x^n \}$ where $y^i = y^i(x^1,...,x^n)$ and $x^i = x^i(y^1,...,y^n)$ for each $i \in \{1,...,...
0
votes
1
answer
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:
$$
\dfrac{\mathbf{V}^T \cdot \mathbf{V}}{\mathbf{V}^T : \mathbf{V}} + \dfrac{\mathbf{V} \cdot \mathbf{V}^T}{\mathbf{V} : \mathbf{V}^T} = \dfrac{\mathbf{D}^2}{\left \| \...
user1012020
1
vote
2
answers
116
views
Why is $\delta\mathbf{u}\cdot \mathrm{div}(\mathbf{\sigma}) = -\mathrm{grad}(\delta\mathbf{u}) \mathbf{:} \mathbf{\sigma}$?
Context is from this deal.ii tutorial. Screenshot of the relevant part is below.
I don't get the transformation from $\delta\mathbf{u}\cdot \mathrm{div}(\mathbf{\sigma})$ to $-\mathrm{grad}(\delta\...
- 866
3
votes
1
answer
6k
views
Calculating the Divergence of a Tensor
I am working through a fluid dynamics paper and came across this equation:
$$ \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 ...}.$$
However, what does ...
- 13.4k
0
votes
0
answers
62
views
Transformations of curvilinear coordinate systems
I'm reading a book covering multiple topics in mathematics and physics. There's a chapter on tensors that had a very curious statement. It says "all curvilinear coordinate systems can relate to ...
- 259
2
votes
1
answer
149
views
Extremely complex vector-matrix expression and its differentiation by vector
Given:
$Q=R_z(\psi)R_y(\xi)R_x(\phi)$ - rotation matrix
$\boldsymbol{\theta}=\left[\begin{array}{@{}c@{}} \phi \\ \xi \\ \psi
\end{array} \right]$ - vector of angles
$p=Q\left[\begin{array}{@{}...
- 731
0
votes
3
answers
362
views
Deriving product rule for divergence of a product of scalar and vector function in tensor notation
On page-94 of the 4th edition in the international version of Griffith's Electrodynamic, the following identity is used:
$$ \int \left[ V(\nabla \cdot \vec{E} ) + \vec{E} \cdot \nabla V \right]dV= \...
0
votes
1
answer
379
views
$C_{ij}=T_{ijklmn} D_{kl} D_{mn}$ 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
answers
404
views
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 $$L(x,y)=f(x_0,y_0)+\nabla f ...
- 33
0
votes
1
answer
42
views
Levi-Civta symbol question
$$\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
1
vote
1
answer
1k
views
Partial derivatives in tensor/index notation!
For my continuum mechanics class, I'm tasked with finding $\nabla u$, $u$ being $u = b\frac{x}{|x|^3}$. Here, $b$ is a scalar constant.
Attempt at the solution:
I rewrite $\frac{1}{|x|^3}$ as $(x_i^2)^...
- 11
2
votes
1
answer
427
views
Proof of expression for Christoffel symbols of the first kind, $[i , j k] = {\bf e}_i \cdot \frac{\partial {\bf e}_j}{\partial x^k}$
On page 155 of Vector and Tensor Analysis with Applications, by A.I Borishenko and I.E. Tarapov, the authors assert that,
$$\frac{\partial {\bf e}_j}{\partial x^k} = \left\{ i \atop j \; k \right\} {\...
- 1,261
1
vote
0
answers
80
views
Calculate the angle between vectors in equation. When does the conductivity tensor component take the form $\sigma_{ab} = \bar \sigmaδ_{ab}$?
In a certain anisotropic conductive material, the relationship between the current density $\vec j$ and
the electric field $\vec E$ is given by: $$\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\...
- 23
1
vote
1
answer
1k
views
How should I calculate fourth order tensor times second order tensor?
Let's say I have two second-order tensors
${\mathbf{S}} = {S_{ij}}{{\mathbf{e}}_i} \otimes {{\mathbf{e}}_j}$
and
${\mathbf{T}} = {T_{ij}}{{\mathbf{e}}_i} \otimes {{\mathbf{e}}_j}$
. Then, I know
${\...
- 11
0
votes
1
answer
149
views
Tensor Calculus Notation
In tensor notation, we know the following is true for general vectors:
$$
\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C}) = A_i\epsilon_{ijk}B_jC_k = -B_j\epsilon_{jik}A_iC_k
$$
However, if we try and ...
- 133