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.