Advanced calculus is a more advanced version of multivariable calculus. Advanced topics in calculus include multivariable limits, partial derivatives, inverse and implicit function theorems, double and triple integrals, vector calculus, divergence theorem and stokes theorem, advanced series, and power series.
Differentiation is one of the major concepts of calculus apart from integration. It is the derivative or rate of change of a function with respect to the independent variable.
The fundamentals of calculus include topics like limits, differential calculus, integral calculus, multivariable calculus. The two primary operations in calculus are differentiation and integration.
Integration is one of the two major calculus topics along with differentiation. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Multivariable calculus is study of calculus in one variable to multiple variables. The typical operations involved are limits and continuity, partial differentiation, and multiple integration. Major applications are in regression analysis, in finance by quantitative analyst, in engineering and social science to study and build high dimensional systems and exhibit deterministic nature.
Real analysis is a branch of mathematical analysis that studies about real numbers, sequences and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes concepts of limits, convergence, continuity, and measure theory.
Vector calculus also called as vector analysis deals with differentiation and integration of the vector field in three-dimensional Euclidean space. It deals with quantities that have both magnitude and direction.