In this article, we show a list of courses to learn advanced mathematics. Although, We offer tips and techniques that can help you overcome almost any of your math hurdles. In fact, We give courses and solved exercises on general algebra, (groups, rings, and fields; linear algebra, vector spaces, dimension, matrices, endomorphisms, eigenvalues, stables vector spaces; real analysis, Cauchy sequences, continuous and uniformly continuous functions, differential calculus, series, and Fourier analysis.
For more details on advanced math, we refer to AMS.
Introduction to general algebra: learn advanced mathematics courses
In this part, we give posts on group theory, fields, and rings as well as on the ring of polynomials
- Group theory: course and exercises with solutions
- Fields and rings problems with detailed solutions
- Exercises with solutions on polynomials
- Some algebra exercises for training
Linear algebra exercises and problems
By linear algebra, we mean the study of vector spaces, linear maps (transformations), and the computation of matrices.
- Vector spaces and dimension
- Matrices calculus
- Linear transformation and the associated matrices
- Trace of Matrices
- Eigenvalues and eigenvectors
- Hyperplanes
- Direct sum of subspaces: facts and examples
Introduction to calculus and real analysis
In this section, we start by discussing the properties of the set of real numbers. This will help to fully understand the concepts of limits and continuity for real functions. An important class of continuous functions is the class of uniformly continuous functions (in particular the Lipshitzien and Holderien functions). In addition, differential calculus is very important because this concept is involved in the study of ordinary and partial differential equations.
- Set of real numbers
- Limits and continuity of real functions
- Simple exercises on the limit of functions
- Recurring sequences
- Tips on sequences convergence
- Cauchy sequences
- Uniformly continuous functions
- Differentiability of functions of one real variable
- Convex functions
- Riemann integrals
- Functions defined by integrals
- Improper integrals
- Parametric integrals
- Convergence of series
- Differential calculus for functions of several variables
More advanced math courses and exercises
We offer more advanced math lessons and solved problems. Such background is useful for undergraduate mathematics students.
- Fourier transform definition, properties, and applications
- Cauchy-Lipschitz theorem for differential equations
- Stability of nonlinear ordinary differential equations
- Peano’s theorem gives the existence of ODE solutions
- The maximal solution to Cauchy problems
- Solve heat equation using Fourier series
- An application of the open mapping theorem
- Differential calculus in Banach spaces
- The optimal step gradient method and applications
