Introductory Calculus Lecture Notes

Practical Information

• Course Structure: 16 lectures, meet twice a week on Mondays and Wednesdays at 10 AM.
• Lecture Notes: Available online (written by Kath Wilkins).
• Problem Sheets: 8 problem sheets, first two are online, covered in 4 tutorials.
• Recommended Reading: Mathematical Methods in Physical Sciences by Mary Boas.

Syllabus Overview

1. Differential Equations (7-8 lectures)

   • Ordinary Differential Equations (ODEs)
   • Partial Differential Equations (PDEs)
   • Techniques for solving and practical examples from physical sciences.

2. Integrals (3 lectures)

   • Line and double integrals
   • Applications: arc lengths, areas of regions and surfaces.

3. 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 for optimization.

Importance and Interconnections

• Directly Useful For: Multivariable calculus, dynamics, PDEs, and analysis courses.
• Mandatory Course: Essential for understanding further options in applied mathematics (e.g., differential equations, fluid dynamics).

Introduction to Differential Equations

• Definition: An equation involving an independent variable (x) and a dependent function (y) along with its derivatives.
• Order of Differential Equation: Defined by the highest derivative present.
  • Example: Simplest ODE is dy/dx = f(x), solvable by direct integration.
• Examples from Physical Sciences:
  • Mechanics: Newton's second law translates to differential equations involving acceleration and velocity.
  • Electrical Circuits: RLC circuits yield second-order differential equations relating voltage, current, and charge.
  • Exercise: Write a differential equation for radioactive decay.

Review of Integration Techniques

• Integration by Parts: Derived from the product rule for derivatives.
  • Formula: ∫u dv = uv - ∫v du

Example 1: ∫ x² sin(x) dx

• Choose u = x², dv = sin(x) dx
• Result: x² (-cos(x)) + 2 ∫ x cos(x) dx

Example 2: ∫ (2x - 1) ln(x) dx

• Choose u = ln(x), dv = (2x - 1) dx
• Result involves long division and substitution.

Solving Differential Equations

• Simplest ODEs: Solvable directly through integration.
• Separable Differential Equations:
  • Form: dy/dx = a(x) b(y)
  • Can be separated and integrated.

Example: Separable DE

• Given: x(y² - 1) + y(x² - 1) dy/dx = 0
• Separate variables to solve: ∫(y/(y² - 1) dy) = -∫(x/(x² - 1) dx).

Conclusion

• Important to master integration techniques for solving differential equations.
• Care should be taken when dividing by variables (e.g., ensuring they are non-zero).
• Next lecture will continue with more differential equations.