Free online courseTensors for Beginners: Learn Tensor Calculus Fundamentals

Course Description

This free online course helps you build a clear, usable understanding of tensor calculus fundamentals without getting lost in heavy formalism. If you have met vectors in calculus or physics and wondered why tensors appear in topics like changing coordinates, curved spaces, or advanced mechanics, this course connects the motivation to practical definitions that actually work when you compute. You will learn to recognize what stays the same under a change of basis and what changes, which is the central idea behind tensors.

Instead of treating tensors as mysterious arrays, you develop intuition step by step: starting from coordinate descriptions, then moving through vectors and their transformation rules, and on to covectors as objects that eat vectors and return numbers. That perspective makes later ideas feel natural, because you can see how dual bases, components, and transformations fit together. Along the way, you practice distinguishing forward and backward transformations so you can reliably translate between old and new coordinate systems.

As the course progresses, tensors appear as structured combinations of vector and covector behavior. You explore linear maps as (1,1)-tensors and understand how their components transform, which clarifies when a matrix representation is tied to a specific basis. You then meet bilinear forms and the metric tensor, learning why they matter for computing lengths and angles correctly in non-orthonormal coordinates and for moving between upstairs and downstairs indices through raising and lowering operations.

You will also gain a conceptual handle on tensor products, how they differ from the Kronecker product in purpose and meaning, and why higher-type tensors can become ambiguous if you only think in terms of symbols rather than multilinear maps. Short exercises throughout reinforce the transformation rules and definitions so you finish with confidence reading index notation, checking whether an expression is coordinate-consistent, and setting yourself up for applications in multivariable calculus, differential geometry, physics, computer graphics, and machine learning.