Free online course Calculus 3

About the free online course

Strengthen the calculus skills that power physics, engineering, computer graphics, data-driven modeling, and advanced mathematics. This free online course takes you beyond single-variable techniques and helps you think in 3D and higher dimensions, where vectors, surfaces, and vector fields become the language for describing real systems. Instead of treating topics as disconnected formulas, you will learn how geometric intuition and algebra work together to solve meaningful problems.

You will begin by developing a solid grasp of vectors in three-dimensional coordinate systems, including operations that reveal structure in space such as dot and cross products. From there, you will connect lines, planes, and common surfaces to a geometric toolkit that makes spatial reasoning faster and more reliable. As the course moves into vector functions, you will learn to describe motion with parameterizations and interpret derivatives as velocity and acceleration, building the bridge between calculus and dynamics.

The course then expands into multivariable functions, where limits, continuity, and partial derivatives help you analyze how outputs change when several inputs vary at once. You will learn how differentials support approximation, how the multivariable chain rule links composed systems, and how gradients and directional derivatives explain steepest ascent in a precise way. With this foundation, you will be able to form tangent planes, normal lines, and optimize functions of two variables, including constrained optimization using Lagrange multipliers.

Integration becomes more powerful as you progress to double and triple integrals, including strategies for setting up iterated integrals and choosing coordinate systems that simplify geometry. You will learn how polar, cylindrical, and spherical coordinates can turn complicated regions into manageable ones, and how change of variables with the Jacobian supports flexible modeling. Applications like mass, moments, and volume reinforce why these methods matter.

Finally, you will work with vector fields and the calculus of circulation and flux: divergence, curl, and line integrals, along with the key theorems that connect local behavior to global results. With videos and practice questions throughout, you will build the conceptual clarity and problem-solving habits needed for exams, future coursework, and technical careers.