Introductory Calculus Course Overview

Aug 31, 2024

Introductory Calculus Lecture Notes

Practical Information

  • Course Structure: 16 lectures total.
  • Schedule: Lectures on Mondays and Wednesdays at 10 AM.
  • Problem Sheets: 8 sheets total; first two are online.
  • Tutorials: 4 tutorials in college covering problem sheets.
  • Lecture Notes: Available online (authored by Kath Wilkins).
  • Recommended Textbook: Mary Boas' Mathematical Methods in Physical Sciences.
    • Concise with examples from physics and engineering.

Syllabus Overview

First Half: Differential Equations

  • Duration: 7-8 lectures.
  • Types:
    • Ordinary Differential Equations (ODEs)
    • Partial Differential Equations (PDEs)
  • Focus: Techniques for solving differential equations, includes educated guesses.

Second Half: Integration

  • Topics:
    • Line and double integrals.
    • Calculating arc lengths and areas.
  • Duration: Approximately 3 lectures.

Third Component: Multivariable Calculus

  • Topics:
    • Functions of two variables.
    • Surfaces, gradients, normal vectors.
    • Taylor's theorem in multiple variables.
    • Critical points and Lagrange multipliers for optimization.

Course Interconnections

  • Related Courses: Directly useful for multivariable calculus, dynamics, and analysis.
  • Important groundwork for future courses in applied mathematics, such as fluid dynamics and wave equations.

Differential Equations Introduction

Ordinary Differential Equations (ODEs)

  • Definition: Involves an independent variable (x) and a dependent variable (y) with derivatives.
  • Example Equation:
    • Simplest form: [ \frac{dy}{dx} = f(x) ]
    • Solvable by direct integration.

Examples from Physical Sciences

  1. Newton's Second Law:
    • Force = mass * acceleration, relates to differential equations through acceleration.
  2. Electrical Circuits:
    • RLC Circuit example leading to a second-order differential equation.
    • Kirchhoff’s law leads to relationships between voltage and current.

Exercise

  • Write a differential equation for radioactive decay proportional to the remaining number of atoms.

Integration Techniques

Integration by Parts

  • Formula: Derived from the product rule for derivatives.
    • Indefinite integral version:
      [ \int f g' , dx = f g - \int f' g , dx ]
  • Examples:
    • [ \int x^2 \sin(x) , dx ]
    • More complex examples requiring substitutions and recognizing logarithmic forms.

Recursive Formulas

  • Integration of Powers of Cosine:
    • Establish a recursive formula for [ \int \cos^n(x) , dx ].
  • Base Cases:
    • Compute base cases (e.g., [ I_0 \text{ and } I_1 ]).
  • General Recursive Formula:
    • Establishes a pattern for calculating higher powers.

Important Points to Remember

  • Techniques learned in this course will prepare students for various applications in mathematics and physical sciences.
  • Mastering integration techniques is crucial for solving differential equations.
  • Pay attention to conditions under which certain operations (like division) are valid to avoid losing solutions.

Conclusion

  • The first lecture introduces the basics of differential equations and integration techniques.
  • Next lecture will continue exploring differential equations.