Introductory Calculus Lecture Notes

Practical Information

• Course structure: 16 lectures, 2 per week (Mondays and Wednesdays at 10 am)
• Lecturer: Dan Ciubotaru
• Lecture notes and reading list available online
• Recommended textbook: Mathematical Methods in Physical Sciences by Mary Boas
• 8 problem sheets to be covered in 4 tutorials (2 problem sheets online already)

Syllabus Overview

First Half: Differential Equations (7-8 Lectures)

• Types of Differential Equations:
  • Ordinary Differential Equations (ODEs)
  • Partial Differential Equations (PDEs)
• Focus on techniques for solving equations and understanding their applications in science and engineering.

Second Half: Integration (3 Lectures)

• Topics include:
  • Line and double integrals
  • Computing arc lengths and areas
  • Introduction to multivariable calculus

Key Concepts in Multivariable Calculus

• Surfaces, gradient, and normal vectors
• Taylor's theorem in two variables
• Critical points and optimization (Lagrange multipliers)

Importance of Introductory Calculus

• Prepares students for:
  • Multivariable calculus
  • Dynamics and PDEs
  • Applied mathematics options in Part A
  • Analysis courses

Introduction to Differential Equations

Definition of ODE

• An equation involving an independent variable (x) and a dependent variable (y) with derivatives:
  • Example: dy/dx = f(x)
• Order of the ODE is determined by the highest derivative present.

Examples of Differential Equations

1. Newton's Second Law (Mechanics):
   • F = ma (a is the acceleration, which is the derivative of velocity)
   • Can lead to second-order ODEs (e.g., a = d²r/dt²)
2. Electrical Circuits (RLC Circuit):
   • Components: Resistor (R), Inductor (L), Capacitor (C)
   • Differential equation relating current (I) and charge (Q):
     • L(d²Q/dt²) + R(dQ/dt) + (1/C)Q = V
   • This is a second-order, inhomogeneous differential equation.
3. Radioactive Decay:
   • Exercise: Write a differential equation describing the decay rate of a radioactive substance.

Review of Integration Techniques

Integration by Parts

• Derived from the Product Rule of differentiation:
  • ∫f(x)g'(x)dx = f(x)g(x) - ∫f'(x)g(x)dx
• Example: Integrating x²sin(x)
  • Steps involve choosing f and g based on their derivatives.

Practice Examples:

1. Integrating 2x - 1 times ln(x² + 1)
   • Use long division and substitution techniques.
2. Finding recursive formulas:
   • Example: Integrating cosⁿ(x)dx, leads to a recursive formula I(n).
   • Understand how to obtain values of I(0) and I(1).

Types of Differential Equations

Separable Differential Equations

• Form: dy/dx = a(x)b(y)
• Can be separated and integrated:
  • 1/b(y) dy = a(x) dx

Example Problem

• Given a specific separable equation, demonstrate the steps to separate and solve it.

Conclusion

• Emphasis on careful handling of differential equations, particularly when dividing by functions that may be zero.
• Reminder of integration techniques and practice.

Upcoming Topics

• More discussion on differential equations in the next lecture.