All Questions
6
questions
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 ...
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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
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
0
votes
1
answer
119
views
Index notation: associative property
I have given the following term, that I can write in index notation as following:
$$ (\mathbf{a} \cdot \nabla)\mathbf{b} = a_j \partial_j b_i$$
Now I can exchange the order and get
$$ \partial_j b_i ...
- 163
1
vote
2
answers
670
views
divergence of gradient of scalar function in tensor form
I found simple expression in tensor notation for a divergence of product vector and gradient of scalar function:
$$\operatorname{div}(\mathbf{j}) = 0 \text{, where } \mathbf{j} = \mathbf{m}\times \...
- 29
0
votes
0
answers
1k
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Examples of Tensor Transformation Law
Let $T_{\mu\nu}$ be a rank $(0,2)$ tensor, $V^\mu$ a vector, and $U_\mu$ a covector. Using the definition of tensors based on the tensor transformation law, determine whether each of the following is ...
- 603