The Complete Physics Companion

The advanced, calculus-based companion to The Complete Physics Crash Course

Introduction: Why This Book Exists

Most physics courses hand you a formula, tell you to plug in numbers, and move on. That approach gets you through an exam, but it leaves the formula feeling like a magic spell instead of a discovered fact. This book takes the opposite approach. Every law of physics in these pages is derived — built up from a smaller set of assumptions using the tools of calculus, so that by the end you understand not just what the equation says but why it has to be true.

This is the advanced, calculus-aware edition. It assumes you are comfortable with single-variable differentiation and integration (derivatives of polynomials, exponentials, and trig functions; definite and indefinite integrals; the idea of a separable differential equation) and with basic vectors and trigonometry. If calculus is new to you, this is the wrong place to start — go work through The Complete Physics Crash Course first, the algebra-based companion volume that covers the same physical world without derivatives or integrals, then come back here once the notation feels familiar. Nothing here is harder physics than the Crash Course; it is the same physics, argued from first principles instead of asserted.

Across eighteen chapters this book derives kinematics from the definition of a derivative, Newton's laws as differential equations, the work-energy theorem from calculus substitution, the wave equation from a free-body diagram on an infinitesimal string segment, Gauss's law from a flux integral, the Schrödinger equation's simplest exact solution, and the Lorentz factor from a light clock. It closes with special relativity, general relativity, and an honest tour of the genuinely unsolved problems at the edge of current research — quantum gravity, dark energy, and the rest — clearly labeled as open, not settled.

You will find over 160 solved practice problems, more than 50 fully worked examples with real step-by-step mathematics, a hands-on laboratory chapter with real error-propagation calculus and a working Python data-fitting script, and a 30-question cumulative final exam with a complete, worked answer key. Two appendices at the back collect every physical constant and every derived formula used in the book into a single quick-reference map, organized by chapter, for when you need to look something up without re-reading the derivation that produced it. Keep a pencil nearby. Derive before you look at the answer — that is where the learning actually happens.

A note on notation. This book writes derivatives as d/dt, dx/dt, or with primes; integrals with ∫; square roots with √; and uses standard Greek letters (π, θ, ω, λ, μ, ρ, Δ, ħ for h/2π, Ω, φ, σ) throughout, matching the notation you will meet in any university-level physics or engineering text. Numerical constants (G, c, h, k_B, e, ε₀, μ₀) use standard CODATA/SI values, listed the first time each appears, and collected in full in Appendix A.

How to use this book. Each chapter follows the same shape: one or more labelled "Derivation" blocks build a result from a smaller set of assumptions, two or three "Worked Example" blocks apply that result to a fully solved numeric problem, and a set of Practice Problems (with answers, but no intermediate working shown) lets you test whether the derivation actually transferred. Work them in that order, and resist the pull to skip straight to the boxed final formula: the formula is the least useful part of a derivation to memorize, because it is the one part you can always look up in Appendix B, while the method of getting there is what generalizes to a problem this book has never shown you. Before checking any answer, commit to a number on paper — a wrong answer you can diagnose teaches far more than a right one you only recognized. When a derivation genuinely does not click on first reading, the most reliable fix is almost never re-reading the same paragraph more slowly; it is closing the book and attempting the derivation from a blank page, using only the physical assumption it starts from (a force law, a boundary condition, a postulate about the speed of light) as a prompt. If you can rebuild the argument from that one sentence, you have actually learned it; if you cannot, you now know precisely which single step needs another look, rather than having a vague sense that "Chapter 9" is somehow unclear.

Part I — Foundations and Classical Mechanics

Chapters 1 through 6 build the calculus toolkit and apply it to the physics of single particles and simple systems: kinematics, Newton's laws, energy, momentum, and oscillation. Every later part of this book — fields, waves, quantum mechanics, relativity — reuses these same six chapters' techniques (a derivative set to zero, a differential equation solved and checked, a conserved quantity tracked through a process) applied to progressively less familiar physical situations, so working through these chapters thoroughly by hand pays off for the rest of the book.

Chapter 1: The Calculus of Physical Quantities

Physics is the science of quantities that change, and calculus is the mathematics of change. A measurement is a number with a unit; a model keeps only the variables that matter for the question being asked. The single biggest upgrade this book offers over an algebra-based course is this: instead of memorizing that "velocity is the slope of a position-time graph," you will treat velocity as the honest derivative of position, and every kinematic formula you have ever memorized will fall out of that one definition.

Derivation: velocity and acceleration as derivatives. Let x(t) describe an object's position along a line. Over a small time interval Δt, the average velocity is Δx/Δt. As Δt shrinks toward zero, this ratio approaches the instantaneous velocity:

v(t) = lim(Δt→0) Δx/Δt = dx/dt

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