The Tensor Confusion: Why Nobody Agrees What a Tensor Is

Computer scientists say tensors are multidimensional arrays. Physicists say tensors are geometric objects that obey transformation laws. Mathematicians say tensors are multilinear maps. How can all of these definitions be talking about the same thing? In this video we begin a journey through the different ways tensors are understood across mathematics, physics, engineering and computer science. We start with tensors as multidimensional arrays, explore the physicist's viewpoint based on transformation laws, and then develop the modern mathematical interpretation of tensors as multilinear maps. Topics covered: • Tensors as multidimensional arrays • Covariant and contravariant components • Tensor transformation laws • Scalars, vectors and covectors • Dual spaces • Multilinear maps • Tensor notation and components • Why different communities define tensors differently This is Part 1 of a series. In the next videos we will develop the tensor product construction, quotient spaces, the Universal Property, and see how these viewpoints are ultimately unified. Chapters: 00:00 Introduction 01:21 Tensors as Arrays 03:25 Physicist's Viewpoint & Essentials Background 10:12 How Different Objects Transform 21:20 Tensor Transformation Laws 23:40 Mathematician's Viewpoint: Multilinear Maps 30:04 Tensor Transformation Laws from Multilinear Maps #Tensor #TensorCalculus #LinearAlgebra #Mathematics #Physics #MultilinearAlgebra #MathEducation #TensorProduct #DifferentialGeometry #STEM #Engineering #Manim