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Maths 2 - Sets, Rings, Fields & Groups

A Bit About Proofs & Logic

The main strategies in proofs are by:

• Counterexample, where given a general statement $P$, to show that $P$ is true it is necessary to give a general proof; but to show that $P$ is false, we have to give one specific instance in which it fails.

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Entropy

Entropy in a quantum information context can be said to represent the amount of uncertainty in our knowledge of a system. A completely mixed state has maximum uncertainty in the sense that it is a complete classical ensemble of multiple possibilities that are equally possible, and thus we can say nothing about which outcome is likely. A pure state however is something that we have absolute knowledge of. We know that it is in that state. Note that a pure state can be a superposition. This does not contribute to its uncertainty in terms of entropy. A superposition and its probability amplitudes can change depending on the measurement basis we choose. That is a manifestation of quantum randomness and not uncertainty in the state of the system.

Von Neumann Entropy

The Von Neumann entropy of a system with a density matrix $\rho$ is defined as,

$$S(\rho)=-\text{Tr}(\rho\log_2\rho)$$

where the logarithm is taken in base 2. The logarithm of a matrix is constructed by taking the log of the eigenva

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Probability & States

A system usually has several different states it can inhabit. These states are called microstates. Microstates that are indistinguishable from each other are clubbed under a single macrostate, and the number of microstates that make up a macrostate is called the multiplicity of that macrostate.

For eg. if we have 100 coins, there is only one way in which every coin will be heads up. i.e. the macrostate of all coins being heads up has a multiplicity of 1. However, the macrostate of one coin being heads up and the rest tails up has a multiplicity of 100 because any one of the 100 coins could be heads up, and each of these represents a different microstate.

If we assume that every microstate is equally probable, a system is most likely to eventually end up in the macrostate with the highest multiplicity i.e. the most probable macrostate. The probability of this occurring is given by:

$$\frac{\Omega_{\text{max}}}{\Omega_{\text{total}}}$$

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The Electromagnetic Field and Potentials

In classical electrodynamics, the fundamental entities are the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$. These fields are mathematically derived from a scalar potential $\phi$ (the electrostatic equivalent of classical voltage potential) and a vector potential $\mathbf{A}$. The fields are defined as:

$$\mathbf{B} = \nabla \times \mathbf{A}$$

$$\mathbf{E} = -\nabla\phi - \frac{\partial \mathbf{A}}{\partial t}$$

E & B as the 4-curl of the 4-vector potential

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Symmetries

From Susskind's Theoretical Minimum Book

Classical Mechanics: Symmetries & Conservation Laws

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1. General Conservation (Noether's Theorem)

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Tensors

The whole story in five lines

1. Tensors are invariant; their components transform — upper indices by B, lower by F (F, B inverse).
2. Vectors (contravariant, upper) and covectors (covariant, lower) are the building blocks; a covector is a stack of lines and α(v) counts piercings.
3. Combine p vectors and q covectors with ⊗ to build any (p,q)-tensor; transform each factor and the rules fall out for free.
4. The metric g_ij = e_i·e_j measures length/angle and rescues Pythagoras outside orthonormal bases.
5. g lowers (♭) and g⁻¹ raises (♯) indices, linking vectors and covectors — but v^i ≠ v_i unless the basis is orthonormal.

What is a tensor: Imagine a car heading on a straight highway, towards a left turn that it wants to take. Imagine that the distance of the car is measured along the highway as x, starting from the left turn and increasing towards the car's present position at time 0. Imagine the y-axis as starting from the left turn's join with the highway

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Barbara Ryden — Introduction to Cosmology (2nd ed., 2016) — Revision Notes

How to read these notes

• Coverage: Chapters 3–12 plus the Epilogue
• Every bullet ends with a citation like [R 4 81] = Ryden, Ch 4, PDF page 81.
• Formulas relevant to a bullet are included inline, tagged with the book's own equation number, e.g. (Eq. 4.20) — the same numbering used in the formulas sheet, which remains the master list.

Quick reference: ★ Essentials Cheat Sheet, Ch 2-6 Formulas, Ch 7-12 Formulas, Ch 3-7 Notes

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Barbara Ryden — Introduction to Cosmology (2nd ed., 2016) — Revision Notes

How to read these notes

• Coverage: Chapters 3–12 plus the Epilogue
• Every bullet ends with a citation like [R 4 81] = Ryden, Ch 4, PDF page 81.
• Formulas relevant to a bullet are included inline, tagged with the book's own equation number, e.g. (Eq. 4.20) — the same numbering used in the formulas sheet, which remains the master list.

Quick reference: ★ Essentials Cheat Sheet, Ch 2-6 Formulas, Ch 7-12 Formulas, Ch 8-12 Notes

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Barbara Ryden — Introduction to Cosmology (2nd ed., 2016) — All Formulas Ch 7-12

How to read this list

• Equation numbers (N.m) are the book's own numbering (stable across printings).
• Page numbers refer to this PDF's page index, not the printed Cambridge page numbers (the file is a calibre EPUB→PDF conversion, so the two differ).

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Contents

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Barbara Ryden — Introduction to Cosmology (2nd ed., 2016) — All Formulas

How to read this list

• Equation numbers (N.m) are the book's own numbering (stable across printings).
• Page numbers refer to this PDF's page index, not the printed Cambridge page numbers (the file is a calibre EPUB→PDF conversion, so the two differ).

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Contents