Newton and Leibniz revolutionized mathematics with calculus. Newton's fluxions focused on rates of change and motion, using dot notation. He applied his method to physics problems like planetary motion.

Leibniz's approach used infinitesimals and differentials, introducing the familiar d/dx notation. His more algebraic style gained popularity in Europe. Both methods tackled continuous change and laid the groundwork for modern calculus.

Newton's Fluxional Calculus

Foundational Concepts of Fluxional Calculus

• Fluxions represent rates of change of continuously varying quantities
• Newton conceptualized fluxions as velocities of moving points generating geometric curves
• Fluents denote the time-dependent quantities themselves, changing continuously over time
• Method of ultimate ratios utilized to determine instantaneous rates of change
  • Involves finding the limit of the ratio of two quantities as they approach zero
  • Avoids the logical issues associated with infinitesimals

Notation and Mathematical Approach

• Newton's dot notation used to represent fluxions (time derivatives)
  • ẋ denotes the fluxion (rate of change) of x
  • ẍ represents the second fluxion (acceleration) of x
• Contrasts with Leibniz's d/dx notation for derivatives
• Newton's approach focused on geometric and kinematic interpretations
  • Emphasized motion and continuous change in physical quantities
• Applied fluxional calculus to solve problems in physics and mechanics (planetary motion)

Leibniz's Differential Calculus

Fundamental Concepts and Infinitesimals

• Differentials represent infinitesimally small changes in variables
  • dx signifies an infinitesimal change in x
  • dy represents the corresponding infinitesimal change in y
• Infinitesimals conceptualized as quantities smaller than any finite number but greater than zero
• Leibniz's approach based on algebraic manipulation of these infinitesimal quantities
• Infinitesimal calculus provided a framework for analyzing continuous change
  • Allowed for precise calculations of rates of change and areas under curves

Continuity and Mathematical Foundations

• Continuity played a crucial role in Leibniz's differential calculus
  • Functions considered continuous if their graphs could be drawn without lifting the pen
• Leibniz's calculus built on the concept of the characteristic triangle
  • Infinitesimal right triangle used to approximate curves locally
• Developed the fundamental theorem of calculus, linking differentiation and integration
• Introduced integral notation ∫ representing the sum of infinitesimal rectangles
• Leibniz's approach more algebraic and symbolic compared to Newton's geometric focus
  • Led to wider adoption and development of calculus in continental Europe