Introductory Calculus Concepts and Techniques

Sep 6, 2024

Introductory Calculus Lecture

Course Overview

  • Lecturer: Dan Chibotaru
  • Number of Lectures: 16
  • Schedule: Mondays and Wednesdays at 10 AM
  • Materials: Lecture notes by Kath Wilkins available online
  • Reading List: Online - Recommended book: "Mathematical Methods in Physical Sciences" by Mary Boas

Course Structure

  • Differential Equations:
    • Ordinary Differential Equations (ODEs)
    • Partial Differential Equations (PDEs)
  • Integrals:
    • Line and Double Integrals
    • Arc lengths and area computations
  • Multivariable Calculus:
    • Functions in two variables
    • Surfaces, gradients, normal vectors
    • Taylor's theorem in two variables
    • Optimization with Lagrange multipliers

Relevance to Other Courses

  • Interaction with multivariable calculus, dynamics, series, and PDEs
  • Foundation for rigorous proofs in Analysis (particularly Analysis II)

Differential Equations in Physical Sciences

  • Ordinary Differential Equations (ODEs): Function of x (independent variable) and its derivatives
    • Example: dy/dx = f(x), solve by integration
  • Applications:
    • Mechanics: Newton's second law, acceleration as a derivative of velocity
    • Electrical Circuits: RLC circuit equations
      • Kirchhoff's law
      • Voltage equations involving resistance, inductance, and capacitance

Integration Techniques

  • Integration by Parts:
    • Derived from the product rule: (f \cdot g' = f'g + fg')
    • Example 1: ( \int x^2 \sin x , dx )
    • Example 2: ( \int (2x - 1) \ln(x^2 + 1) , dx )
  • Reduction Formula: Recursive formula for integrals e.g., ( \int \cos^n(x) , dx )

Solving Differential Equations

  • Separable Differential Equations: e.g., dy/dx = a(x) * b(y)
    • Separate variables and integrate
    • Example problem: Solve (x(y^2 - 1) + y(x^2 - 1) \frac{dy}{dx} = 0)

Exercises

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

Conclusion

  • Focus of the next lecture will be more on differential equations.

Additional Tips

  • Practice integration and differential equations
  • Be cautious when dividing by terms that could be zero in separable equations

Ensure to review and practice integration by parts, substitution, and solving differential equations as they form the basis of this course. This lecture sets the groundwork for more complex topics in calculus and applied mathematics.