Published in Military Embedded Systems
Over time, the definition of a tensor has varied across communities from mathematics to quantum physics. Lately, it has joined the machine learning community’s lexicon. If you search the web for the definition of a tensor, you will likely be overwhelmed by the varying explanations and heated discussions. In 1900, Gregorio Ricci Curbastro and his student Tullio Levi-Civita first published their theory of tensor calculus, which is also known as absolute differential calculus.
The importance of tensor calculus became apparent in 1915 when physicist Albert Einstein revealed that he had found it indispensable for the gravitational field equations used in his theory of general relativity. From March to May of 1915, Einstein and Levi-Civita wrote to each other, their correspondence filled with complex mathematical equations, proofs, and counterproofs. All 11 letters that Einstein wrote to Levi-Civita have survived, while only one of Levi-Civita’s letters still exists. To honor Levi-Civita, the mathematical permutation symbol, εijk, used in tensor calculus, is known today as the Levi-Civita symbol.
One way to understand the importance of tensor calculus is to consider geometric complications when drawing right angles. If you are developing a system that uses the flat-earth model, you can draw right angles using the Pythagorean Theorem. The limits of the Pythagorean Theorem become clear when you try to draw a right angle on a spherical surface. In this case, the Pythagorean Theorem no longer works. It’s here that the metric tensor comes to the rescue. It generalizes coordinates and geometries so that distance can be measured in any given space. The magic of tensors comes from their special transformational properties that enable them to describe the same physics in all reference frames.