Introductory Calculus

Practical Information

• Course Structure

  • 16 lectures, notes online by Cath Wilkins
  • Lectures: Mondays and Wednesdays at 10am
  • 8 problem sheets covered in 4 tutorials

• Lecture Details

  • Lecturer: Dan Ciubotaru
  • Reading list online
  • Recommended book: Mary Boas's Mathematical Methods in Physical Sciences

Syllabus Overview

Differential Equations

• First half (7-8 lectures) on:
  • Ordinary Differential Equations (ODEs)
  • Partial Differential Equations (PDEs)
• Techniques for solving DEs

Integrals

• Line and Double Integrals (about 3 lectures)
  • Useful for computing arc lengths and areas

Multivariable Calculus

• Introduction to calculus of functions in two variables
• Topics include:
  • Surfaces, gradients, normal vectors
  • Taylor’s theorem in two variables
  • Critical points
  • Lagrange multipliers

Interaction with Other Courses

• Useful for:
  • Multivariable calculus
  • Dynamics
  • PDEs (next term)
  • Fourier series and PDEs
  • Analysis, especially Analysis II

Practical Applications of DEs

Examples from Physical Sciences

Mechanics

• Newton’s second law: F = ma
• Acceleration as derivative: a = dv/dt
• Example: Second-order DE from displacement

Electrical Circuits

• Example: RLC circuit
  • Components: Resistor (R), Inductor (L), Capacitor (C), Voltage source (V)
  • Differential equations involving current I(t) and charge Q(t)
  • Kirchoff’s law
  • Second order DE: L * d²Q/dt² + R * dQ/dt + (1/C) * Q = V*

Radioactive Decay

• Exercise: Rate of radioactive substance decay proportional to remaining atoms

Integration Techniques

Review of Integration by Parts

• Product rule: f(x)g’(x) = (fg)’ - f’g
• Indefinite integral: ∫f(x)g’(x)dx = f(x)g(x) - ∫f’(x)g(x)dx
• Definite integrals include limits of integration

Examples

1. ∫x²sin(x)dx
2. ∫(2x - 1)ln(x² + 1)dx
3. Recursive formula for ∫cosⁿ(x)dx

Separable Differential Equations

• Form: dy/dx = a(x) * b(y)
• Example to solve: x(y² - 1) + y(x² - 1) dy/dx = 0*

Additional Notes

• Practice integration by parts and other integration techniques
• Solve simplest cases of DEs with direct integration or separation of variables
• Example exercises given during lecture

Next Steps

• Review lecture examples and try similar problems
• Prepare for next lecture on differential equations