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Questions tagged [index-notation]

For questions about index notations, e.g. abstract index notation, Einstein summation convention, topics in introductive tensor calculus, Levi-Civita Symbol, Kronecker Delta symbol, proofs of vector calculus identities or fluid dynamics formulae using index notation.

For questions about index notations, including abstract index notation, Einstein summation convention, topics in introductive tensor calculus, Levi-Civita Symbol, Kronecker Delta symbol, proofs of vector calculus identities or fluid dynamics formulae using index notation.

488 questions
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Transposed matrix index notation confusion

In an attempt to understand tensors I am reading this document. In page 15 we have $A^{\mu}_{\alpha}a^{\alpha}A^{\mu}_{\beta}b^{\beta}=(A^T)^{\mu}_{\alpha}A^{\mu}_{\beta}a^{\alpha}b^{\beta}$. I can't see where that transposition comes from or why…
XxS4NN4SxX
  • 15
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1 answer

How to write summation of squared divergence terms in index summation notation?

Sorry if this has already been asked before, but it's really difficult to try and explain the problem in words. Anyways, I want to express the following: $$\phi = \left(\frac{\partial u_1}{\partial x_1}\right)^2 + \left(\frac{\partial u_2}{\partial…
James Wright
  • 207
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1 answer

Find $\bigcap \mathcal A^c$ and $\bigcup \mathcal A^c$ with the proof.

Let $A_n =\{x \in \Bbb R :-\frac {1}{n} \lt x \lt \frac{1}{n}\}$,$n \in \Bbb N$ and define the indexed family $\mathcal A^c = \{ A_{n}^{c} :n \in \Bbb N \}$. Find $\bigcap \mathcal A^c$ and $\bigcup \mathcal A^c$ with the proof. I have no idea how…
Maggie
  • 369
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1 answer

Index Notation Question

I wanted to know if someone could help me with this, please. This is my progress so far with the question: $N = 2\times 5^3\times x^4$ $N = 250x^4$ $N^2 = (250x^4)^2$ $N^2 = 62500x^8$ $5N^2 = 5(62500x^8)$ $5N^2 = 312500x^8$ Thanks
user476859
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1 answer

Expressing the magnitude of a vector difference in indicial notation

I'm trying to express the following relation in indicial notation $$ |\vec{u} - \vec{v}_p| \, . $$ The only way I found out is replacing the difference above by $$ \vec{u} - \vec{v}_p = \vec{v}_r \, , $$ then one can write $$ |\vec{u} - \vec{v}_p| =…
Diego Venturi
  • 13
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1 answer

Show with index notation that $||\nabla \times \underline{u}||^2=||\nabla \underline{u}||^2 - \mathbf{Tr}[(\nabla \underline{u})^2]$

So far I have: $$||\nabla\times\underline{u}||^2 = \left[(\nabla\times\underline{u})\right]_i\left[(\nabla\times\underline{u})\right]_k =\left[\varepsilon_{ijk}\frac{\partial u_k}{\partial x_j}\right]\left[\varepsilon_{kmn}\frac{\partial…
MRT
  • 603
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1 answer

Simplify indices expression

Neither I nor my maths teacher can simplify this expression to get the answer in our textbook. Can you show us how, or is the textbook wrong? The expression The textbook claims the simplified expression is:$$4/21$$
MarkPorter
  • 11
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Practicing Index Notation proofs

I am practicing proving things with index notation in order to get a good grasp of the way it works. One of the problems that I came up against is $\nabla(\frac{1}{2}v^2) = \vec{v}\times(\nabla\times\vec{v})+(\vec{v} \cdot \nabla)\vec{v} $ You can…
Coolcrab
  • 111
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1 answer

Einstein Notation Summing Order

When summing an expression using Einstein notation we start with the leftmost index? For example in that case of $$a^{i}_jb^j_kc^k_l$$ We start with $k$ or…
gbox
  • 12,867
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A Problem with Index Notation

Suppose we have a set of numbers $T_{ij}=B_{ij}+a_i$ and another set $A_{ij}$. Now, according to index notation, $A_{ij}T_{ij}=A_{ij}B_{ij}+A_{ij}a_i$. The $1^{st}$ term in the RHS is a summation over both indices $i$ and $j$ while the $2^{nd}$ is…
Tofi
  • 277
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2 answers

Solving for an unknown variable in an index without using logarithms

I have come across the following equation in chapter 1 of a book that is testing applications of index laws. Logarithms are not covered until chapter 17. I can't figure out how to solve for $t$ without using logarithms. $$ 1000 = 20 \times …
user757957
-1
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4 answers

What is $10^{40}$ as a number?

What is $10^{40}$? Every time I google this question I get $1\mathrm{e}\!+\!40$ but I don’t understand this so what is it as a number?
Ty Jury
  • 19
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