# Tagged Questions

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### Einstein Summation - does the following equality hold: $a_{ij} x_i y_j = a_{ij} y_i x_j$

Does equality hold when $x_i = y_i$ and $x_j=y_j,$ and $i, j = 1, ..., n$.
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### Einstein Summation: How do I show $a_{ij} (x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j$?

Einstein Summation: How do I show $a_{ij} (x_i + y_j) \not= a_{ij}x_i + a_{ij}y_j$?
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### What does this operator $\odot$ mean

I read this about the second fundamental form in Wikipedia and I’ve no idea what does $\odot$ mean? Does anybody know? $$II=-dN\cdot dP=\omega^3_1\odot\omega^1+\omega^3_2\odot\omega^2$$
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### What is the notation for pull-back and push-forward of an exponential map?

So there is a nice notation for a one-parameter group of transformations $\Phi_t$ corresponding to its infinitesimal generator $\boldsymbol X$: $$\Phi_t = \exp \left(t \boldsymbol X \right)$$ But ...
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### Space of Alternating $k$-Tensors Notation

I will be taking a Differential Geometry class in the Fall, so I decided to get somewhat of a head start by going through Spivak's "Calculus on Manifolds." Before reading, though, I saw the Addenda at ...
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I was surprised to notice that gradient of function and Levi-Civita connection have the same notation, i.e. nabla sign $\nabla$. Moreover, extending any connection on tensors, one let it be ...
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### Accepted symbol (or way of writing) “A is a subset of B or B is a subset of A”

I am looking for a concise way to write the statement "$A$ is a subset of $B$ or $B$ is a subset of $A$". The context is the Grassmannian and two elements $A,B\in G_k(\mathbb R^n)$ in it. The two ...
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### Integral notation from cartesian from polar coordinates

Given an integral $$I=\int\limits_{\mathbb{R}^n} \cdot \; dx,$$ we can introduce polar coordinates, such that $$I=\int\limits_{\Bbb S^{n-1}} \cdot \; d\theta.$$ Another way to express the latter one ...
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### Doubts with differential geometry notation in Frankel

This is from Frankel's The Geometry of Physics: Problem 2.3(2) Consider the tangent bundle to a manifold $M$. Show that under a change of coordinates in $M$, $\partial/\partial q$ depends ...
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### Which set is this $I_p(p)\cdot \Gamma(E)$?

Let $\pi:E\rightarrow M$ be a smooth vector bundle and $p\in M$. Consider $$I_p(M)=\{f\in C^\infty(M): f(p)=0\},$$ and $\Gamma(E)$ the $C^\infty(M)$-module of smooth sections over $E$. Notice $I_p(M)$ ...
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### Problem with notation: Laplacian on a manifold

In the Aubin's book "Nonlinear analysis on manifolds" the Laplacian operator on functions on some smooth manifold is defined by the formula $$\Delta = -\nabla^\gamma\nabla_\gamma,$$ where ...
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### Notation: subscript vs. superscript for coordinate vector fields

Some books write the coordinate vector fields with a subscript as $$\frac{\partial}{\partial x_i}$$ while some write it with a superscript as $$\frac{\partial}{\partial x^i}.$$ Is there a conceptual ...
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### Standard notation for isometry group?

Let $M$ be a (semi) Riemannian manifold, is there a standard notation for the group of isometries on $M$? I would think $\mathrm{ISO}(M)$ would be appropriate, but I've never encountered a dedicated ...
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### Why is the Jacobian matrix the transpose of what I would think it'd be/usefully be (total derivative is a synonym) (EDIT: I was a total wally)

I'm sorry this isn't a yes/no/am-I-right question but I seriously cannot see why the Jacobian/total derivative matrix is what it is? I am also using it as LaTeX practice (for maths) hence the barely ...
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### “Product” bundle notation.

Let $\newcommand{\Spin}{\operatorname{Spin}}M$ and $M'$ be two manifolds, equipped with a principal $\Spin_n$ and $\Spin_{n'}$ bundle called $P$ and $P'$, respectively. Then there is an induced ...
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### what is $C^{-\infty}(\mathbb{R})$

Thanks in advance. what is $C^{-\infty}(\mathbb{R})$? Is that the same as the "distribution" defined in differential geometry? It would be helpful if someone can describe it in another way ...
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### Notation to work with vector-valued differential forms

What it the standard notation used while working with vector-valued differential forms? I tried using abstract index notation, for example denoting a $1$-form valued $2$-form as $P_{i[bc]}$, but I'm ...
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### Notation and naming for two operations with $p$-form valued $n$-forms

While trying to answer my other question I found I never heard about vector-valued differential forms. I've been searching for them in various mathematical physics books, but didn't get too much. I'm ...
I saw papers saying something like "let $\gamma:S^1 \times [0,T] \to \mathbb{R}^2$ parametrise a curve. The second interval above just makes it time dependent, but why parametrise (for fixed time) the ...