Introductory Calculus Lecture Notes

Practical Information

• Instructor: Dan Chibotaru
• Lecture Schedule: Mondays and Wednesdays at 10 AM
• Total Lectures: 16
• Lecture Notes: Available online (written by Kath Wilkins)
• Problem Sheets: 8 problem sheets, first two available online
• Tutorials: 4 hours of tutorials in college
• Recommended Reading: Mary Boas' "Mathematical Methods in Physical Sciences"

Syllabus Overview

• Focus Areas:
  • Differential Equations (7-8 lectures)
    • Ordinary Differential Equations (ODEs)
    • Partial Differential Equations (PDEs)
  • Line and Double Integrals (3 lectures)
    • Application: Compute arc lengths and areas
  • Multivariable Calculus (final lectures)
    • Topics: Surfaces, gradients, Taylor's theorem, critical points, Lagrange multipliers for optimization

Differential Equations Introduction

• Definition: An ODE involves an independent variable (x) and a dependent variable (y) along with its derivatives.
• Order: The order is determined by the highest derivative present.
• Example ODE:
  • Simplest form:
    • dy/dx = f(x)
    • Solution: y = ∫ f(x) dx

Examples from Physical Sciences

• Newton's Second Law:
  • F = m * a, where a = dV/dt, resulting in second-order ODE.
• Electrical Circuits (RLC Circuit):
  • Kirchhoff's law leads to a second-order differential equation involving current (I(t)) and charge (Q(t)).

Integration Techniques

• Integration by Parts:
  • Derived from the product rule of differentiation.
  • Formula:
    • ∫ u dv = uv - ∫ v du
  • Example:
    • ∫ x² sin x dx

Recursive Integration Example

• Integral of cosⁿ(x) leads to a recursive formula:
  • iₙ = (1/n) cosⁿ⁻¹(x) sin(x) + (n-1)/n iₙ⁻²

General Solution to Separable Differential Equations

• Example:
  • dy/dx = f(x) * g(y)
  • Method: Separate variables and integrate each side.
• Final Note:
  • Care needed when dividing by functions that could be zero.

Conclusion

• The first lecture laid the groundwork for differential equations and integration techniques.
• Next lecture will continue with more examples and applications of differential equations.