Video 01 - Why Tensor Calculus

Jim Fowler introduces a series of videos on tensor calculus, expressing his passion for the subject and its significance in mathematics. He aims to share his knowledge to foster appreciation for tensor calculus. The video begins by addressing the fundamental questions about what tensor calculus is, its applications, and its benefits. Fowler emphasizes the strength of tensor calculus in presenting physics equations independently of coordinate choices, which is crucial for scientific modeling. He also discusses the importance of understanding geometric principles and having robust mathematical tools, such as tensor calculus, to analyze physical phenomena.

This paragraph delves into the historical progression of mathematical understanding from basic algebra and Euclidean geometry to the introduction of coordinate systems, leading to analytic geometry. Fowler shares his fascination with the discoveries made in geometry without a coordinate system, exemplified by the Pythagorean theorem. He then contrasts this with the analytical capabilities enabled by coordinate systems, such as calculating midpoints, lengths, areas, and working with curves and surfaces in multiple dimensions. The paragraph also touches on the representation of vectors and operations in differential calculus and vector analysis, highlighting the importance of choosing the right coordinate system to avoid obscuring the geometric nature of problems.

Fowler discusses the principle of performing mathematical analysis at the highest level of abstraction before moving to lower levels when necessary. He uses historical examples involving Leonard Euler, Joseph Lagrange, and Jean Baptista Monge to illustrate how choosing the wrong coordinate system can obscure the geometric significance of solutions. The paragraph emphasizes the importance of recognizing the geometric nature of mathematical objects as invariants and the pitfalls of relying too heavily on specific coordinate systems. Fowler reinforces the idea with a problem involving minimal surfaces and the mean curvature tensor, showing how tensor calculus can simplify complex problems.

This section provides an overview of the major contributors to the development of tensor calculus. Bernard Riemann, Elwin Bruno Christophel, Gregorio Ricci-Curbastro, Tulio Levy-Cevita, Marcel Grossman, and Albert Einstein are highlighted for their significant contributions to the field. The paragraph traces the evolution of tensor calculus from its roots in differential geometry to its formal establishment by Ricci-Curbastro and its application in Einstein's general theory of relativity. The narrative underscores the impact of these mathematicians on the advancement of tensor calculus and its role in modern physics.

Fowler outlines the prerequisites for understanding tensor calculus, suggesting a strong foundation in algebra, trigonometry, calculus, analytical geometry, vector analysis, differential equations, and linear algebra. He clarifies that while a background in these areas is beneficial, it is not strictly necessary, as the focus is on mathematics rather than science or physics. He also recommends a book by Grenfell on tensor analysis, which he will be using as a guide for the video series, and emphasizes that the series is about mathematics and not about science, making it accessible to a broader audience.

The final paragraph wraps up the video with a central takeaway: the essence of tensor calculus is to acknowledge the need for coordinate systems while avoiding commitment to a specific system for as long as possible. Tensor calculus provides algorithms for constructing analytical expressions that are valid across all coordinate systems, thus maintaining the geometric integrity of solutions. Fowler ends with a quote from Grenfell's book that encapsulates the rationale behind tensor calculus, setting the stage for the next video, which will review the properties of Euclidean space.