References (with Comments from the Author)
This book's peculiar algebraic approach was built by combining the insights of the following pioneers.
1. Daniel Fleisch, A Student's Guide to Vectors and Tensors, Cambridge University Press (2011)
Comment: A careful guide to the traditional approach. A well-regarded introduction that treats traditional vector analysis and tensor analysis component calculation with extreme care. The best book for acquiring the power to break through head-on the jungle of Christoffel symbols and index manipulation—the route this book tries to bypass via differential forms.
2. David Bachman, A Geometric Approach to Differential Forms, Birkhäuser (2nd ed. 2011)
Comment: A geometric introduction to differential forms. A good book that explains differential forms not from axioms but from geometric intuition such as "measuring shadows." Much of this book's intuition that "$dx$ is a measuring device" comes from Bachman, but here computational closure is prioritized and that intuition is adopted by dropping it into matrix algebra.
3. Harley Flanders, Differential Forms with Applications to the Physical Sciences, Dover Publications (1989/1963)
Comment: A classic of physical-mathematical application. A bible-like masterpiece for applying differential forms to physics. The rigorous operational rules of the exterior derivative $d$ and Hodge star $\ast$ are condensed here. Because it starts from an axiomatic introduction, self-study is somewhat hard for beginners, but readers who finish this book should be able to wield its powerful calculations fully.
4. William L. Burke, Applied Differential Geometry, Cambridge University Press (1985) William L. Burke, Div, Grad, Curl are Dead (Unfinished Manuscript)
Comment: The structural pillar of this book—the philosophy of "delaying the metric." The most important literature that fixed the skeleton of this book. Burke advocated the educational philosophy of "putting the metric later and later into the course." This philosophy, which made explicit how much can be said with the exterior derivative $d$ before introducing the metric, supports the book's structure up to the delayed introduction of the metric and Hodge star in Chapter 6.
5. Leonard Susskind & Art Friedman, Quantum Mechanics: The Theoretical Minimum, Basic Books (2014)
Comment: Intuition for matrix representation. The intuition of translating abstract $1$-forms (dual space) into the physicist's mother tongue—"lay column vectors on their sides as row vectors and take matrix products"—was strongly influenced by the operational feel of bra–ket notation in quantum mechanics.
6. Chris Doran & Anthony Lasenby, Geometric Algebra for Physicists, Cambridge University Press (2003)
Comment: One unification beyond this book. An entry to geometric algebra (Clifford algebra) for modern physics. One sees the landscape where "inner product" and "outer product," thoroughly separated in this book, are integrated from the start as a single noncommutative product. Readers who know the pain of separation can feel how powerfully liberating that unification is.