All Questions
Tagged with vector-analysis tensors
143
questions
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112
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tensor identity
The tensor $t$ is defined in terms of a scalar $\varrho$ and the two vectors $v$ and $u$ (and derivatives) as follows:
$$t_{i j}:=\varrho v_i u_j-\frac{1}{2} \varrho \varepsilon_{i lm} v_l (\...
0
votes
1
answer
56
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$\nabla\cdot(f\vec{g})=f\nabla\vec{g}+\vec{g}\cdot\nabla f$ using Levi-Civita
I need to prove the following equality: $\nabla\cdot(f\vec{g})=f\nabla\vec{g}+\vec{g}\cdot\nabla f$.
I know that proving this equality using the properties of $\nabla$ and $(\cdot)$ is easy, what the ...
- 135
2
votes
1
answer
74
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geometric interpretation on covariant derivateve in curvilinear coordinates.
I'm having trouble understanding what does the covariant derivative do in a coordinate system where we have changing basis vectors. I always thought it was giving us the change in coordinates while ...
- 379
25
votes
7
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How would you explain a tensor to a computer scientist?
How would you explain a tensor to a computer scientist? My friend, who studies computer science, recently asked me what a tensor was. I study physics, and I tried my best to explain what a tensor is, ...
2
votes
1
answer
66
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Differential surface element and nabla operator
If we have the vector field $\vec{u}=\vec{A}\times \vec{v}$, where $\vec{A}=\text{const.}$ and we integrate over some closed curve, by using Stokes' theorem we get:
$$
\begin{align}
\oint_{\partial S}...
- 379
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1
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44
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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
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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
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votes
2
answers
118
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What is the gradient in this skewed coordinate system?
Consider two bases $e_i$ and $f_i$ of $\mathbb{R}^2$ defined by:
$$\begin{aligned}
(f_1,f_2) &= (e_1,e_2)\cdot F\\
\begin{pmatrix}1&-1\\1&1\end{pmatrix} &= \begin{pmatrix}1&...
- 88
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51
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Intuition about the divergence of a vector field in non-orthogonal basis
My textbook defines the divergence of a vector field in a non orthogonal constant basis the following way:
$$div(\vec{u})=\vec{a}^i\cdot\frac{\partial \vec{a}_ku^k}{\partial x^i}=\frac{\partial u^i}{\...
- 379
1
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answers
45
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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.
$$ \int\; \boldsymbol{A} \times (\boldsymbol{\nabla} a ) = \pm \int (\boldsymbol{\nabla} \times \boldsymbol{...
- 11
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answers
51
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Differential operators of tensor fields. Hamilton operator
The very first thing my textbook says is that the Hamilton operator is defined as:
$$\vec{\nabla}=\vec{a}^i\nabla_i$$
Where $\nabla_i$ is the covariant derivative and " $\vec{a}^i$ is the ...
- 379
2
votes
2
answers
93
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Derivation or intuition on the covariant derivative for higher rank tensors
So the derivation in my textbook for the covariant derivative of a vector field $\vec{u}$ in curvilinear coordinates $\xi^k$ is the following:
$$\frac{\partial \vec{u}}{\partial\xi^j}=\frac{\partial (...
- 379
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answers
25
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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
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0
answers
92
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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
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1
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59
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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 ...
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