Very few people have the ability to work in areas of advanced mathematics and therefore never get to experience the magnificent beauty of its topics, among which the following are typically included:

- Logic
- Methods of Proof
- Set Theory
- Cartesian Products and Functions
- Relations
- Cardinality
- Number Theory
- Rings and Integral Domains
- Limits and the Real Numbers
- Second and Higher-Order Linear Ordinary Differential Equations
- Systems of Ordinary Differential Equations: Phase Plane: Qualitative Methods
- Series Solutions of Ordinary Differential Equations: Special Functions
- Laplace Transforms
- Linear Algebra: Matrix Eigenvalue Problems
- Vector Differential Calculus
- Vector Integral Calculus
- Fourier Series, Integrals, and Transforms
- Complex Integration
- Power, Taylor, and Laurent Series: Residue Integration
- Complex Analysis and Potential Theory
- Numerics for Ordinary and Partial Differential Equations
- Unconstrained Optimization: Linear Programming
- Graphs: Combinatorial Optimization
- Mathematical Statistics
- Vector Analysis
- Multiple Line and Surface Integrals and Integral Theorems
- Complex Variables and Conformal Mapping
- Complex Inversion Formula for Laplace Transforms

To keep up to date with the latest work in advanced math, you must read the Journal of Advanced Mathematics and Applications. There are also a small number of tutorial websites that cover advanced math topics, one of which has advanced calculus tutorials.

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