XylyXylyX: What is a Tensor?
Forty lessons building tensors from the axioms of a vector space
About this series
Most introductions define a tensor by how its components transform and move on. XylyXylyX does the opposite: 40 lessons that start from the axioms of a vector space and build the tensor product, the metric, the covariant and Lie derivatives, the exterior algebra and Hodge duality in order, defining each object before using it. The transformation law arrives as a consequence rather than a definition.
It pairs naturally with EigenChris's series, which is the more visual treatment: watch that one for geometric intuition and this one for the algebraic scaffolding underneath. Both lead into the written tensor calculus course and general relativity.
Lesson 1 — Elementary vector spaces
Lesson 2 — How to make a map.
Lesson 3 — Dual Spaces
Lesson 4 — Cartesian Products
Lesson 5 — Tensor Products
Lesson 6 — Tensor Product Spaces
Lesson 7 — Rank of a TSP
Lesson 8 — Linear Transformations
Lesson 9 — TPS transformations
About these lectures
The series is worked at the pace of a real course and assumes linear algebra but little else. Lessons 19 and 20 step sideways into the algebraic structures — modules, rings, algebras — that the rest of the material quietly relies on, and lessons 37 and 38 spend two full lectures on what a differential form actually looks like, which most treatments leave to the imagination.
All content is publicly available on YouTube and belongs to its creator. Watch the full series on the original playlist.