Free online courseTensors for Beginners: Learn Tensor Calculus Fundamentals

Course Description

Build a solid, beginner-friendly understanding of tensor calculus with this free online course in Basic Education focused on Calculus. It starts from the motivation behind tensors and carefully develops the core language needed to work confidently with coordinates and transformations.

You will move from clear definitions of vectors and their transformation rules to the dual viewpoint through covectors, including how components behave under a change of basis. From there, the course connects these ideas to linear maps and the rules that keep calculations consistent across coordinate systems.

As the foundations settle, you will explore the metric tensor and how it enables precise geometric measurement, along with bilinear forms and how they relate to combinations of covectors. The course then unifies the framework by showing how tensors can be built from general mixtures of vectors and covectors, and how tensor product spaces formalize these constructions.

To complete the toolkit, you will learn how index notation reflects the underlying structure, including raising and lowering indices and the intuition behind sharp and flat operators. Along the way, you will also clarify the difference between the tensor product and the Kronecker product, helping you avoid common confusion when moving between abstract definitions and matrix representations.