Introductory Calculus Lecture Notes

Course Information

• Lecturer: Dan Chibotaru
• Lectures: 16 lectures, meeting on Mondays and Wednesdays at 10 AM.
• Materials:
  • Lecture notes (written by Kath Wilkins) and reading list available online.
  • Recommended book: Mathematical Methods in Physical Sciences by Mary Boas.
• Problem Sheets:
  • 8 problem sheets, first two available online, discussed in 4 tutorials at college.

Course Syllabus Overview

• First Half: Differential Equations (7-8 lectures)
  • Types: Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs).
  • Techniques and educated guesses for solving.
• Second Half: Integration
  • Topics: Line integrals, double integrals, calculus of functions in two variables.
  • Useful for computing arc lengths and areas.
• Final Topics: Multivariable Calculus
  • Surfaces, gradients, normal vectors, Taylor's theorem, critical points, Lagrange multipliers.
  • Interaction with Dynamics, PDEs, Analysis, and applied mathematics options.

Differential Equations

Definition and Examples

• Ordinary Differential Equation (ODE):
  • Involves an independent variable (x), a dependent variable (y), and the derivatives of y.
  • Example: Simplest form is (\frac{dy}{dx} = f(x)).
  • Solvable by direct integration: (y = \int f(x) , dx).
• Example from Mechanics:
  • Newton's second law (Force = Mass × Acceleration) is a differential equation.
  • Acceleration is a derivative of velocity, leading to second-order ODEs.
• Example from Electrical Circuits:
  • RLC circuit produces a second-order, inhomogeneous differential equation.

Exercise

• Find a differential equation for the rate of radioactive decay being proportional to remaining atoms.

Integration Review

Techniques and Examples

• Integration by Parts

  • Comes from the product rule for derivatives.
  • Formula: (\int u , dv = uv - \int v , du).
  • Example: (\int x^2 \sin x , dx) solved by parts.

• Examples:

  1. (\int x^2 \sin x , dx)
  2. (\int (2x - 1) \ln(x^2 + 1) , dx)
  3. Recursive reduction formula: ( I_n = \int \cos^n x , dx)

Solving Differential Equations

Separable Differential Equations

• Form: (\frac{dy}{dx} = a(x)b(y))
• Method:
  • Separate variables: (\int \frac{1}{b(y)} , dy = \int a(x) , dx)

Example

• Solve (x(y^2 - 1) + y(x^2 - 1) \frac{dy}{dx} = 0)
  • Separate variables, integrate both sides, solve for y.
  • Consider special cases when dividing by expressions, ensure solutions are not missed.

Conclusion

• Be meticulous in separating variables and check for cases when terms may be zero.
• Practice integration techniques and solving differential equations to strengthen understanding.
• Next lecture will continue with more on differential equations.