Comprehensive University-Level Mathematics Courses Online for Free

Embarking on a self-guided journey through university-level mathematics is a commendable endeavor. Fortunately, several reputable platforms offer extensive, free resources that mirror a bachelor's degree program in mathematics. Below is a detailed guide to structured learning paths covering essential topics such as calculus, linear algebra, differential equations, abstract algebra, and real analysis. This guide includes recommended sequencing, estimated completion times, and links to high-quality resources featuring lecture videos, problem sets, and practice exercises.

1. MIT OpenCourseWare (MIT OCW)

MIT OpenCourseWare is a premier resource offering free access to a vast array of courses from the Massachusetts Institute of Technology. The mathematics courses available here are comprehensive and closely follow the curriculum of a typical bachelor's program.

Course Sequence and Details

1. Single Variable Calculus (18.01SC)

   • Topics: Limits, derivatives, integrals, applications of calculus.
   • Resources: Single Variable Calculus (18.01SC) – Includes lecture videos, problem sets, and exams with solutions.
   • Estimated Time: 3-4 months.

2. Multivariable Calculus (18.02SC)

   • Topics: Partial derivatives, multiple integrals, vector calculus.
   • Resources: Multivariable Calculus (18.02SC) – Includes comprehensive lecture materials and assessments.
   • Estimated Time: 3-4 months.

3. Linear Algebra (18.06)

   • Topics: Matrix theory, vector spaces, determinants, eigenvalues, and eigenvectors.
   • Instructor: Prof. Gilbert Strang.
   • Resources: Linear Algebra (18.06) – Features detailed lectures, assignments, and exams.
   • Estimated Time: 3-4 months.

4. Differential Equations (18.03)

   • Topics: First-order and second-order differential equations, Laplace transforms, Fourier series.
   • Resources: Differential Equations (18.03) – Provides extensive lecture notes, videos, and problem sets.
   • Estimated Time: 3-4 months.

5. Abstract Algebra (18.701)

   • Topics: Groups, rings, fields, and modules.
   • Resources: Abstract Algebra (18.701) – Includes lecture materials and comprehensive assessments.
   • Estimated Time: 4-5 months.

6. Real Analysis (18.100B)

   • Topics: Metric spaces, sequences, series, continuity, differentiation, integration.
   • Resources: Real Analysis (18.100B) – Offers detailed lecture notes, problem sets, and exams.
   • Estimated Time: 4-5 months.

2. Khan Academy

Khan Academy is renowned for its interactive and beginner-friendly approach to mathematics. It serves as an excellent supplementary resource, especially for reinforcing foundational concepts through practice exercises and instructional videos.

Course Sequence and Details

1. Calculus I, II, III

   • Topics: Single-variable and multivariable calculus.
   • Resources: Khan Academy Calculus – Features video lectures, interactive exercises, and quizzes.

2. Linear Algebra

   • Topics: Matrices, vector spaces, eigenvalues, and eigenvectors.
   • Resources: Khan Academy Linear Algebra – Provides comprehensive video tutorials and practice problems.

3. Differential Equations

   • Topics: First-order and second-order differential equations, Laplace transforms.
   • Resources: Khan Academy Differential Equations – Includes instructional videos and interactive exercises.

Additional Notes:

• Khan Academy is ideal for building intuition and reinforcing concepts.
• It’s recommended to use Khan Academy alongside more rigorous courses like those on MIT OCW.

3. edX and Coursera

edX and Coursera host a plethora of courses from various prestigious universities. These platforms offer structured paths with options for both auditing courses for free and pursuing certifications.

Recommended Courses

• Calculus:

  • Calculus: Single Variable by the University of Pennsylvania on Coursera – Enroll Here.
  • Introduction to Calculus by the University of Sydney on Open Culture – Learn More.

• Linear Algebra:

  • Linear Algebra - Foundations to Frontiers by The University of Texas at Austin on edX – Enroll Here.
  • Linear Algebra by Rice University on Coursera – Learn More.

• Differential Equations:

  • Ordinary Differential Equations and Linear Algebra by Rice University on Coursera – Enroll Here.
  • Differential Equations by the University of Michigan on Coursera – Learn More.

• Real Analysis:

  • Introduction to Probability and Real Analysis by Harvard University on edX – Enroll Here.

• Abstract Algebra:

  • While standalone free courses are scarce, introductory materials can be found through platforms like edX and Coursera by searching specific algebra topics.

Additional Notes:

• These platforms offer both free auditing and paid certification options.
• Courses typically require 4-6 weeks with a commitment of 4-6 hours per week.

4. Supplementary Resources

To enhance your learning experience, consider incorporating these supplementary resources which provide additional explanations, practice problems, and alternative teaching styles.

Paul’s Online Math Notes

Paul’s Online Math Notes is an excellent resource for detailed written explanations and a plethora of practice problems with solutions. It is particularly useful for self-learners who prefer textual content alongside video lectures.

• Calculus: Paul’s Online Math Notes - Calculus
• Differential Equations: Paul’s Online Math Notes - Differential Equations

YouTube Channels

Several YouTube channels offer high-quality mathematics lectures, providing visual and auditory learning opportunities that complement your study materials.

1. 3Blue1Brown

   • Focus: Intuitive explanations of calculus, linear algebra, and more.
   • URL: 3Blue1Brown

2. Professor Leonard

   • Focus: Full-length lectures on calculus, linear algebra, and differential equations.
   • URL: Professor Leonard

3. MathTheBeautiful

   • Focus: Real analysis, abstract algebra, and linear algebra.
   • URL: MathTheBeautiful

OpenStax

OpenStax provides free, peer-reviewed textbooks that are excellent complements to your online courses, offering in-depth coverage of mathematical concepts.

• Calculus Volumes 1, 2, and 3: OpenStax Calculus
• Linear Algebra: OpenStax Linear Algebra

5. Structured Study Plan

To effectively cover all necessary topics and build a solid mathematical foundation, follow the recommended sequencing below. This plan assumes a self-paced approach with a commitment of approximately 8-10 hours of study per week.

Year 1: Foundational Calculus and Linear Algebra

1. Single Variable Calculus (MIT OCW 18.01SC) – 3-4 months.
2. Multivariable Calculus (MIT OCW 18.02SC) – 3-4 months.
3. Linear Algebra (MIT OCW 18.06) – 3-4 months.

Year 2: Advanced Topics and Differential Equations

4. Differential Equations (MIT OCW 18.03) – 3-4 months.
5. Real Analysis (MIT OCW 18.100B) – 4-5 months.
6. Abstract Algebra (MIT OCW 18.701) – 4-5 months.

Year 3: Specialized Topics and Consolidation

7. Advanced Abstract Algebra (MIT OCW 18.702) – 4-5 months.
8. Additional Topics: Explore subjects like topology, complex analysis, or combinatorics through supplementary courses and resources.

Total Estimated Completion Time: Approximately 2-3 years.

6. Additional Tips for Success

• Consistency: Maintain a regular study schedule to ensure steady progress.
• Practice: Regularly solve problem sets and exercises to reinforce your understanding.
• Supplement Learning: Use multiple resources to gain different perspectives on complex topics.
• Engage with the Community: Participate in forums like MathOverflow or Reddit's Math Community to ask questions and engage in discussions.

7. Conclusion

By following the structured paths outlined above and utilizing the recommended resources, you can effectively achieve a comprehensive understanding of university-level mathematics equivalent to a bachelor's degree. The combination of rigorous course materials from MIT OCW, interactive practice from Khan Academy, and supplementary resources like Paul's Online Math Notes and specialized YouTube channels will provide a well-rounded and in-depth mathematical education. Commitment, consistency, and active engagement with the material are key to successfully mastering these subjects.