Contents
Each heading is a link to the corresponding section in its chapter.
Part I: Differential Forms on $\mathbb{R}^3$ (Chapters 1–5)
Chapter 1: What Is $dx$? — A Measuring Device That Eats Vectors, or a Row Vector
- §1.0 The Mathematician’s One Dimension, the Physicist’s One Dimension
- §1.1 Riemann Sums and Matrix Products
- §1.2 The Total Differential $df$ — An Operator That Packs Rates of Change into a Matrix
- §1.3 Leibniz Notation and Algebraic Intuition
- §1.4 Coordinate Transformations — Rebuilding Measuring Devices in New Coordinates
- §1.5 Writing in Another Coordinate System — Exercises
- §1.6 Summary of This Chapter and Outlook Toward the Next
Chapter 2: What Is Area? — The Sign Rule Hidden in Parallelepipeds
- §2.0 Measuring Devices, Area, Volume, and Length
- §2.1 The Limits of Elementary-School Area
- §2.2 The Three Rules an Area-Measuring Device Must Satisfy
- §2.3 An Area-Measuring Device Is an "Antisymmetric Matrix"
- §2.4 Internal Structure of the Area-Measuring Device
- §2.5 Volume-Measuring Devices and Determinants
- §2.6 Summary of This Chapter and Outlook Toward Chapter 3
- Appendix A: Tensor-Product Representation of the Volume-Measuring Device — Full Component Calculation
Chapter 3: What Does It Mean to Integrate? — Count Finite Masses, Then Take the Limit
- §3.0 Measuring Curved Things — Paying Back a Debt from Elementary School
- §3.1 Volume — Three Dimensions, Coefficient 1
- §3.2 Surface Area — The Limit of Two Dimensions, Coefficient 1
- §3.3 Curves — The Limit of One Dimension, Coefficient 1, Revealed
- §3.4 Adding Coefficients — Density Times Geometry
- §3.5 Summary of This Chapter and Outlook Toward the Next
Chapter 4: What Is a Change of Variables? — The Pullback $\Phi^*$: Making Measuring Devices Consistent
- §4.0 Physics Curves; Computation Uses Boxes
- §4.1 Pullback of a 1-Form — Work and the Work-Energy Theorem
- §4.2 Pullback of a 2-Form — Conservation of Angular Momentum and Areal Velocity
- §4.3 Pullback of a 3-Form — Conservation of Mass and Volume Integrals
- §4.4 Properties of the Pullback — What We Have Established So Far
- §4.5 Summary of This Chapter and Outlook Toward Chapter 5, the Exterior Derivative
Chapter 5: What Does It Mean to Differentiate? — The Exterior Derivative $d$: Integral Quantities and Local Laws
- §5.0 The Bridge to Differentiation — Observation Is Integral, Law Is Differential
- §5.1 Revisiting $df$ — Are Differentiation and Integration Inverse Operations?
- §5.2 The "Mismatch" Revealed by a Closed Loop
- §5.3 Dismantling an Infinitesimal Loop — The Mismatch Is Proportional to Area
- §5.4 The Birth of $d$ — A New Measuring Device for Mismatch per Unit Area
- §5.5 The Exterior Derivative of a General $1$-Form — Extension to Three Dimensions
- §5.6 Accumulate It, and Only the Boundary Remains — Stokes' Theorem
- §5.7 The Same Thing One Degree Higher — Exterior Derivative of a $2$-Form and Divergence
- §5.8 $d^2 = 0$ — The Mismatch of a Mismatch Leaves Nothing Behind
- §5.9 Unifying the Exterior Derivative — One Formula, One Rule
- §5.10 From Integrals to Differential Equations — Localizing Physical Laws
- §5.11 Outlook Toward Part II — Foreshadowing the Hodge Star
- Appendix B: Matrix Representation of the Exterior Derivative
- B.1 $0$-form: the $1 \times 3$ row vector of $df$
- B.2 $1$-form: the Jacobian matrix $\mathbf{J}$ of the coefficients
- B.3 $d\omega = \mathbf{J}^T - \mathbf{J}$
- B.4 $2$-form: $d\eta$ and the trace of the Jacobian matrix
- B.5 $d^2 f = 0$ and the Hessian
- Appendix C: Integral Forms of Electromagnetism and Differential Forms
- C.1 Degrees of physical quantities
- C.2 Gauss's law for charge
- C.3 The law of magnetic flux
- C.4 Faraday's law
- C.5 Ampère–Maxwell's law
- C.6 Where a metric is needed
Part II: Vector Analysis (Chapters 6–9)
Chapter 6: The Metric $g$ and the Hodge Star $\ast$ — Summoning the Inner Product and Reversing Degree
- §6.0 The End of Excuses — Releasing the Inner Product
- §6.1 The Inner Product on Parameter Space — The True Nature of the Metric $g$
- §6.2 Converting Between Column Vectors and Row Vectors Using $g$
- §6.3 The Hodge Star $\ast$ — The Correspondence Connecting Two Routes
- §6.4 Examples of the Correspondence — Differential Forms and Vector Analysis
- §6.5 The Types of the Three Operations — grad, curl, div
- §6.6 Toward Vector Analysis
- Appendix D: Array Representation of the Hodge Star
- D.1 $\ast_{1\to2}$ — placing three coefficients into an antisymmetric matrix
- D.2 $\ast_{2\to1}$ — extracting coefficients by the Frobenius product
- D.3 $\ast\ast=\mathrm{id}$ in the $1$-form and $2$-form case
- D.4 Array representation of $\ast_{0\to3}$ and $\ast_{3\to0}$
- D.5 Summary
Chapter 7: Vector Analysis — Enter Nabla
- §7.0 Enter Nabla
- §7.1 Dot Products and Cross Products
- §7.2 $\nabla$ and the Gradient
- §7.3 Divergence
- §7.4 Curl
- §7.5 The Laplacian
- §7.6 Identities
- §7.7 A Formula Collection for Nabla
- §7.8 Integral Theorems — Stokes, Gauss, and Green
- §7.9 Toward Chapter 8 — Do Not Look at the Arrows; Look at the Measuring Devices
Chapter 8: Two Languages — Differentiating Measuring Devices and Differentiating Fields
- §8.0 The Highlight of This Book
- §8.1 Two Differentials, Two Worlds
- §8.2 Completing the Translation Dictionary
- §8.3 Translating Stokes' Theorem
- §8.4 Translating Gauss' Theorem
- §8.5 Why the Two Languages Agree
- §8.6 Curvilinear Coordinates and the Two Routes
Chapter 9: In Practice — Build the Dictionary and Solve Hard Problems
- §9.0 The Central Tools of This Book Are Now in Place
- §9.1 Building the Dictionary on the Spot — A Mechanical Procedure
- §9.2 Cylindrical Coordinates $(r,\theta,z)$
- §9.3 Spherical Coordinates $(\rho,\theta,\phi)$
- §9.4 A Hard Problem in Calculus — The Vector Laplacian in Spherical Coordinates
- §9.5 A Hard Problem in Electromagnetism — Divergence of the Point-Charge Electric Field
- §9.6 The Dictionary Ends; the Journey Continues
Part III: Extensions and Unification (Chapters 10–12)
Chapter 10: Maxwell's Equations — Beyond Beauty
- §10.0 The Obligatory Maxwell Chapter
- §10.1 Maxwell's Equations — Two Equations
- §10.2 The Electromagnetic Field $F$ and Fixing Sign Conventions
- §10.3 The Minkowski Metric — From $\mathbb{R}^3$ to Four Dimensions
- §10.4 Writing Out $dF=0$ in Full
- §10.5 $\ast F$ and the Remaining Two Equations
- §10.6 Constructing the Potential — Starting from $F=-d\mathcal{A}$
- Appendix E: Slice-Matrix Representation of $d_4F$ and $d_4(\ast_4F)$ — Seeing Maxwell's Equations as $4\times4\times4$ Arrays
- E.1 Basis $3$-forms and their slice matrices — all 16
- E.2 Writing $dF$ with slice matrices
- E.3 Extracting coefficients by the Frobenius product
- E.4 Reading $dF=0$ through the slices
- E.5 Slice representation of $d_4(\ast_4F) = \mu_0(\ast_4\mathcal{J})$
- Appendix F: The Four Equations of Chapter 5 and the Two Equations of Chapter 10
- F.1 $(E,B,J,\rho_{\mathrm e})$ on space
- F.2 The four equations on space
- F.3 Two equations emerge from $d_4F=0$
- F.4 The remaining two emerge from $d_4(\ast_4F)=\mu_0(\ast_4\mathcal J)$
- F.5 Summary
Chapter 11: Toward Curved Spaces — What Lies Beyond This Book
- §11.0 The Position of This Chapter — A Guidepost for Readers Who Want to Look Further Ahead
- §11.1 Manifolds — Starting from $\mathbf{g}(x)$
- §11.2 Riemannian Geometry — The Tensor-Analysis Style
- §11.3 Beyond That