Introductory Calculus Lecture Notes

Practical Information

• Lecturer: Dan Chibotaru
• Class Schedule:
  • Mondays and Wednesdays at 10 AM
  • Special first meeting
• Course Materials:
  • 16 lectures
  • Lecture notes and reading list available online
  • Recommended book: Mathematical Methods in Physical Sciences by Mary Boas

Course Structure

• Differential Equations (first 7-8 lectures):
  • Ordinary Differential Equations (ODEs)
  • Partial Differential Equations (PDEs)
• Integration (about 3 lectures):
  • Line and double integrals
  • Applications in computing arc lengths, areas
• Multivariable Calculus Introduction:
  • Surfaces, gradients, normal vectors
  • Taylor's theorem in two variables
  • Lagrange multipliers for optimization

Related Courses and Applications

• Interactions with multivariable calculus, dynamics, PDEs
• Links to future courses in applied mathematics

Examples of Differential Equations

• Mechanics: Newton's second law, acceleration as differential
• Electrical Circuits: Series RLC circuit
  • Formulated differential equations using Kirchhoff's law

Simplest ODE Example

• dy/dx = f(x) can be solved by direct integration

Integration Techniques

• Integration by Parts:
  • Derived from product rule
  • Example problems involving integration by parts
• Reduction Formula:
  • Recursive formulas for integrals (e.g., cos^n(x))

Solving Differential Equations

• Separable Equations:
  • Example: Separate variables for x and y
  • Integration of both sides
  • Careful with division by terms that can be zero

Important Concepts

• Solving differential equations involves both science and art
• Interplay between calculus, analysis, and applied mathematics

Homework/Exercise

• Write a differential equation for radioactive decay problem

Conclusion

• Review and practice integration techniques
• Next lecture to continue with differential equations