Notes on Calculus Lecture

Introduction to Calculus

• Mathematical study of change
• Two main branches:
  • Differential Calculus: Rate of change, slope as a function
  • Integral Calculus: Determining quantities like areas and volumes under changing conditions

Key Figures in Calculus

• Isaac Newton

  • Known for the laws of motion and gravitation
  • Proposed light is made of particles (carboscales)
  • Developed calculus out of necessity for physics
  • Called his calculus the method of fluxons
  • Defined variables as "fluents" and their velocities as "fluxons"

• Gottfried Leibniz

  • Developed similar calculus concepts independently
  • More mathematical interpretation, focused on sums of infinitesimal distances
  • Notation used today largely attributed to Leibniz

Historical Context of Calculus

• Babylonia

  • Ancient Mesopotamian state (modern Iraq, Syria, Egypt)
  • Capital: Babylon (60 miles from Baghdad)
  • Early intuition on infinite processes, could determine square roots with high precision
  • Used a sexagesimal number system (base 60)
  • Recorded mathematics on clay tablets (cuneiforms)

• Ancient Greek Contributions

  • Eudoxus of Nidus and Archimedes: Proved areas of shapes using the method of exhaustion
    • Split polygons into triangles to find areas
    • Approximated circle areas by inscribing polygons with increasing sides
    • Introduced the concept of limits and infinite processes

• Medieval Indian Contributions

  • Bhaskara II (12th century): Wrote Siddhanta Shadomani, proved that instantaneous speed of a planet is zero at its peak
  • Madhava (14th century): Linked infinite series with geometry and trigonometry
    • Developed series expansions for sine, cosine, arc-tan, and pi
    • Contributions later referenced by Newton and Leibniz, but his works are not preserved

Evolution of Calculus

• Multiple cultures contributed to calculus around the same time
• Newton and Leibniz are credited with laying the foundations, but many others were involved
• Newton's work was driven by physics, while Leibniz focused on mathematical notations
• Both conceived ideas independently in the 1660s, with Leibniz publishing first
• Both are recognized posthumously as inventors of calculus

Conclusion

• The development of calculus was not a singular event but a culmination of knowledge over thousands of years
• Continues to influence modern fields, including machine learning and AI
• Encouragement to subscribe for more content on data sciences and mathematics.