Glossary

• German word for attempt, approach.
• math A rough guess of solution.

• Circular object with rotational symmetry.

• Circular object with rotational and reflective symmetry.

Function whose domain is a family of subsets of a set, and codomain is the extended real number line.

• It is a series \(\sum t_{n} \), such that \(t_{n}=a_{n+1}-a_{n}\) for some sequence \((a_{n})\).
• It is useful because the partial sum cancels down to only two terms. This cancellation technique is called method of differences, or sometimes telescoping.

• Olympiad level counting (Generating functions) - YouTube
• An expression which its series expansion contains the sequence as coefficients.
• This allows one to make a claim about the global properties of the coefficients.

Set of functions between two fixed sets.

Collection of objects that can be unambiguously defined by a property that all its members share.

• A map \( \varphi\colon X/\sim\, \to Y \) is well-defined if all the elements that are equivalent to \( x\in X \) under the equivalence relation \( \sim \) maps to the same element \( y\in Y \).
• And generally, there is no case in which it is not undefined.

• An object \( A \) is positive definite if there's a sense in which: \[ A > 0\quad \text{(positive)} \] for every cases, besides the trivial case in which \( A = 0\ \text{(definite)} \).
• The semidefiniteness also allows \( A=0 \) for non-trivial cases.
• It is similarly defined for negative definite ( \( < \) ) and negative semidefinite ( \( \le \) ).

A set \( A \) is complete, if every Cauchy sequence converges within \( A \).

• The adjective for the edge cases.
• The opposite is well-behaved.

• The condition that gives rise to a qualitatively different object from the rest, the degenerate case.

In a normed space, \[ \Vert\mathbf{a}+\mathbf{b}\Vert \leq \Vert \mathbf{a}\Vert +\Vert\mathbf{b}\Vert. \] It can be proved with the law of cosine within the Euclidean space

Analyze the form of the problem and guess the solution.

• Noticing the pattern.
• Making an educated guess.

• This name I came up with.
• An equation in terms of the components that represents a set of equations one for each index.

• Naming slots with indices

• Einstein Summation Convention, Einstein Summation Notation
• The implicit summation when indices gets paired.
• Covariant indices are lower and contravariant indices are upper. (sometimes ignored)
• The horizontal position of the index can be used to represent the order in which the tensor takes arguments.

• Descending Factorial, Falling Sequential Product, Lower Factorial

• Pochhammer Function, Pochhammer Polynomial, Ascending Factorial, Rising Sequential Product, Upper Factorial

These are my notations

• It does seem a little too unwieldy but, what do I know? it is a complicated operation in the first place.
• \[ D\bigg|_y \]
  • Fix \(y\) while taking derivatives.
  • \[ \frac{\partial}{\partial x}\bigg|_\text{rot}\mathbf{r} \]
  • hold the rotating frame of reference. take it to be invariant.
  • \[ \frac{\partial}{\partial P}\!\!\!\!\!\!\mathop{\bigg|}\limits_{n, V, T} \]
• \[ D\bigg|^y \]
  • Vary only \(y\), that is, take derivative with respect to \(y\)
  • This is to alleviate the conflicting notations \(D_x\) (with respect to \( x \)) and \(D_\mathbf{x}\) (in the direction \( \mathbf{x} \))

• \(\nabla^2\) is mathematically not convincing, \(\Delta\) gets confused with the difference, the spacing of \(|\nabla|^2\) looks horrible.
• Therefore, let's use \(\nabla^{\cdot 2}\).

• \(\mathrm{d}^{\wedge n}\mathbf{r}\). Imaging taking the parameters of \(\mathbf{r}\) and making a corresponding volume form out of it.
• This makes the dimension and the variable that the one is integrating with respect to clear.

• The arguments of a function or the additional structures of an object can be omitted when it's clear—suppressed.
• e.g. \(f(x, y) \rightsquigarrow f\) and \((M, d) \rightsquigarrow M\)
• big list - Suggestions for good notation - MathOverflow
• \(\looparrowright\) \looparrowright: Immersion of smooth manifolds
• \(\hookrightarrow\) \hookrightarrow: Is used for embeddings and inclusions.
• \(G \curvearrowright X\) \curvearrowright: Group action of \(G\) on \(X\)
• \(2^X\) : The power set of \(X\).
  • Also for the homset \(\mathrm{hom}(A, B)\): \(B^A\), and set of functors from \(\mathcal{C}\) to \(\mathcal{D}\): \(\mathcal{D}^\mathcal{C}\)
  • This is reasonable since:
    • \[ (A^B)^C = A^{B\times C} \]
    • \[ A^B \times A^C = A^{B\sqcup C} \]
• \(\displaystyle \binom{X}{k}\): Set of \(k\)-subsets of \(X\)
  • Notice \[ 2^X = \bigcup_{k=0}^{|X|} \binom{X}{k} \]
  • Which is analogous to binomial coefficient