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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
Newton's Second Law
:
Force = mass * acceleration, relates to differential equations through acceleration.
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.
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