Introductory Calculus Lecture Notes

Practical Information

• Lectures: 16 total
• Lecture Notes: Written by Cath Wilkins, available online
• Lecturer: Dan Ciubotaru
• Schedule: Mondays and Wednesdays at 10am
• Problem Sheets: 8 total, first two online
• Tutorials: 4 tutorials, 4 hours in total
• Reading List: Available online
  • Recommended Book: Mathematical Methods in Physical Sciences by Mary Boas

Syllabus Overview

1. Differential Equations

• Approx. 7-8 lectures
• Types:
  • Ordinary Differential Equations (ODEs)
  • Partial Differential Equations (PDEs)
• Techniques: Combination of science and art

2. Line and Double Integrals

• Approx. 3 lectures
• Compute arc lengths, areas of regions, surfaces

3. Calculus of Functions in Two Variables

• Gentle introduction into multivariable calculus
• Topics:
  • Surfaces
  • Gradient
  • Normal Vectors
  • Taylor's Theorem in two variables
  • Critical Points
  • Lagrange Multipliers

Course Interaction

• Directly useful in multivariable calculus and many applied mathematics options such as differential equations, fluid dynamics, etc.
• Interaction with analysis, particularly Analysis II
• Rigorous proofs in analysis next term

Ordinary Differential Equations (ODEs)

• Definition: An equation involving an independent variable (x) and a function of x (y), and their derivatives
• Simple Example:
  • dy/dx = f(x)
  • Solution: y = ∫f(x)dx
• Example from Mechanics: Newton's Second Law (F = ma = m d²r/dt²)
• Example from Electrical Circuits (RLC circuit):
  • L d²Q/dt² + R dQ/dt + Q/C = V(t)
  • Second-order, constant coefficients, inhomogeneous

Integration Techniques

Integration by Parts

• Formula:
  • Indefinite Integral: ∫u dv = uv - ∫v du
  • Definite Integral: ∫[a,b] u dv = [uv]|[a,b] - ∫[a,b] v du
• Example:
  • Integrate ∫x² sin(x) dx
  • Steps: Identify u = x² and dv = sin(x) dx
  • Result: -(x² cos(x) - 2∫x cos(x) dx - 2∫sin x dx)

Integration by Parts for Recursive Formulas

• Example: ∫cos^n(x) dx
  • Reduction formula: I(n) = (1/n) cos^(n-1)(x)sin(x) + ((n-1)/n) I(n-2)
  • Compute base cases I(0) and I(1) for recursive solution

Separable Differential Equations

• Form: dy/dx = a(x) b(y)
• Solution Method: Separate variables and integrate
• Example:
  • Given: xy² + x² - 1 dy/dx = 0
  • Steps: Separate variables and integrate
  • Result: Implicit solution involving y and x