Introductory Calculus - Lecture 1

Jul 22, 2024

Introductory Calculus - Lecture 1

Practical Information

  • Lectures: 16 lectures total
    • Notes available online (written by Cath Wilkins)
    • Lecturer: Dan Ciubotaru
    • Schedule: Mondays and Wednesdays at 10 am
    • 8 problem sheets, first 2 available online
    • 4 tutorials in college (1 hour each)
  • Reading List: Available online
    • Recommended book: Mathematical Methods in Physical Sciences by Mary Boas

Syllabus Overview

  • First Half: Differential Equations (7-8 lectures)
    • Ordinary Differential Equations (ODEs)
    • Partial Differential Equations (PDEs)
  • Second Half: Line and Double Integrals (3 lectures)
    • Compute arc lengths & areas
  • Final Lectures: Calculus of functions in two variables
    • Introduction to multivariable calculus
    • Topics: Surfaces, gradients, normal vectors, Taylor's theorem in two variables, critical points, Lagrange multipliers

Course Utility

  • Interaction with other preliminary courses (e.g., Multivariable Calculus, Dynamics, PDEs)
  • Useful for applied mathematics options in Part A (e.g., differential equations, fluid and waves)
  • Mandatory course

Differential Equations (DEs)

Ordinary Differential Equations (ODEs)

  • Involves independent variable ( x ) and function ( y(x) )
  • Example: ( \frac{dy}{dx} = f(x) )
    • Solved via integration: ( y = \int f(x) dx )
  • Example in Mechanics: Newton's second law
    • Force = mass ( \times ) acceleration (( \frac{d^2r}{dt^2} ))
  • Example in Electrical Circuits: RLC circuit
    • Components: Resistor (R), Inductor (L), Capacitor (C), Voltage source (V)
    • Kirchhoff’s Law: ( V(t) = RI(t) + L \frac{dI}{dt} + \frac{1}{C}Q(t) )
    • Differential equation in ( Q ): ( L \frac{d^2Q}{dt^2} + R \frac{dQ}{dt} + \frac{1}{C}Q = V(t) )

Integration Techniques

  • Integration by Parts: Originates from the product rule
    • Formula: ( \int f(x)g'(x) dx = f(x)g(x) - \int f'(x)g(x) dx )
    • Example: ( \int x^2 \sin(x) dx )
  • Recursive Integration: Involves reduction formulas
    • Example: ( \int \cos^n(x) dx )
    • Derive a recurrence relation to solve higher integrals

Separable Differential Equations

  • Form: ( \frac{dy}{dx} = A(x)B(y) )
  • Solution: ( \int \frac{1}{B(y)} dy = \int A(x) dx )
  • Example: Solve ( x(y^2 - 1) + y(x^2 - 1) \frac{dy}{dx} = 0 )
    • Steps:
      1. Rearrange: ( y(x^2 - 1) \frac{dy}{dx} = -x(y^2 - 1) )
      2. Separate variables: ( \frac{y}{y^2 - 1} dy = - \frac{x}{x^2 - 1} dx )
      3. Integrate both sides: ( \int \frac{y}{y^2 - 1} dy = - \int \frac{x}{x^2 - 1} dx )
      4. Solve the integrals to find the solution implicitly

Conclusion

  • Review integration techniques if necessary
  • Next lecture: More on differential equations