Introduction to Calculus

Practical Information

• Lecturer: Dan Ciubotaru (Call him Dan)
• Class Schedule: Mondays and Wednesdays at 10 AM
• Course Duration: 16 lectures
• Materials:
  • Lecture notes by Cath Wilkins (available online)
  • Reading List:
    • Mary Boas's Mathematical Methods in Physical Sciences
• Assignments: 8 Problem Sheets
  • Available online
  • Covered in 4 tutorials in college

Course Outline

Differential Equations (7-8 lectures)

• Types Covered:
  • Ordinary Differential Equations (ODEs)
  • Partial Differential Equations (PDEs)
• Techniques:
  • Combination of science and art
  • Educated guesses

Line and Double Integrals (3 lectures)

• Applications:
  • Compute arc lengths of curves
  • Compute areas of regions in the plane or surfaces

Calculus of Functions in Two Variables

• Topics Covered:
  • Various surfaces
  • Gradient, normal vectors
  • Taylor's theorem in two variables
  • Critical points and Lagrange multipliers (optimization problems)

Interaction with Other Courses

• Useful in Multivariable Calculus, Dynamics, PDEs, Fourier series and PDEs
• Interacts significantly with Analysis (especially Analysis II)
• Continuation into Part A with many applied mathematics options

Differential Equations in Physical Sciences

Ordinary Differential Equations (ODEs)

• Definition: Involving an independent variable x and a dependent variable y
• Example: Simplest form: dy/dx = f(x)
  • Solved by direct integration: y = ∫f(x)dx

Example from Mechanics (Newton's Second Law)

• Law: F = ma (Force = mass * acceleration)
• Acceleration: Derivative of velocity ( abla v)
  • v = dr/dt; then a = d^2r/dt^2 (second-order DE)*

Example from Electrical Engineering (RLC Circuit)

• Components: Resistor (R), Inductor (L), Capacitor (C), Voltage source (V)
• Relation: Current I(t), Charge Q(t)
  • I = dQ/dt (Kirchhoff's Law)
  • Combine to form second-order DE: L(d^2Q/dt^2) + R(dQ/dt) + (1/C)Q = V

Exercise

• Problem: Rate of radioactive decay proportional to remaining atoms
  • Write a DE to describe this situation

Integration Techniques Review

Integration by Parts

• Formula: ∫u dv = uv - ∫v du

Example: ∫x^2 sin(x) dx

• Solution Steps:
  • Let u = x^2, dv = sin(x) dx
  • Solve iteratively via parts

Example: ∫(2x - 1)ln(x^2 + 1) dx

• Solution Steps:
  • Let u = ln(x^2 + 1), dv = (2x-1) dx
  • Apply integration by parts and simplify

Recursive Integration Example: ∫cos^n(x) dx

• Reduction Formula:
  • I_n = 1/n cos^(n-1)(x) sin(x) + (n-1)/n I_(n-2)_

Computing Integrals

• Base Cases:
  • I_0 = x + C
  • I_1 = sin(x) + C
• Recursive Example:
  • Calculate I_6 using reduction formula

Differential Equation Types

Separable Differential Equations

• Form: dy/dx = a(x)b(y)
• Solution:
  • Separate variables and integrate both sides

Example Problem

• Equation: x(y^2 -1) + y(x^2-1) dy/dx = 0
• Solution Steps:
  • Isolate, separate variables and integrate both sides
  • Solve using known integration techniques