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Mathematics for Scientists and Engineers I - Calculus
Course Code: PH1110/EE1110
Lecturer: Dr. Stephen West
Institution: Department of Physics, Royal Holloway, University of London
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Contents
Introduction
Functions
2.1 Definition of a Function
2.2 Logarithms - a quick revision
2.3 Trigonometric Functions - a quick revision and a little more
2.4 Power Series
2.4.1 Expanding Functions in Power Series - Taylor Series
2.4.2 Approximation Errors in Taylor Series
Differential Calculus
3.1 Continuity of a function
3.2 Differentiability and Differentiation
3.3 Rules of Differentiation
3.3.1 Product Function
3.3.2 Composite Function
3.3.3 Quotient Rule
3.3.4 Logarithmic Differentiation
3.3.5 Leibniz’s Theorem
3.4 Special Points of a Function
3.4.1 Singular Points
3.5 Inverse Functions
3.5.1 Inverse Trigonometric Functions
3.5.2 Inverse Function Rule
3.6 Partial Differentiation
A note on coordinate systems
4.1 Circular Polar Coordinates
4.2 Cylindrical Polars
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- 4.3 Spherical Polars5. Integration- 5.1 Integration From First Principles- 5.2 Integration of Arbitrary Continuous Function- 5.2.1 Properties of Definite Integrals- 5.3 Integration as the inverse of Differentiation- 5.4 Integration By Inspection- 5.5 Integration by Substitution- 5.5.1 Indefinite Integrals- 5.5.2 Solving Differential Equations with Integration- 5.5.3 Definite Integrals- 5.6 Integration by Parts- 5.7 Differentiation of an Integral Containing a Parameter- 5.8 An Aside on Hyperbolic Functions- 5.8.1 Identities of Hyperbolic Functions- 5.8.2 Inverses of Hyperbolic Functions- 5.8.3 Calculus of Hyperbolic Functions- 5.9 Hyperbolic Functions In Integration by Substitution6. Differential Equations- 6.1 Solving ODEs- 6.2 First order ODE- 6.2.1 Separable first order ODE- 6.2.2 Almost Separable first order ODEs- 6.2.3 Homogeneous first order ODEs- 6.2.4 Homogeneous apart from a constant- 6.2.5 Looks homogeneous but actually separable with a change of variable- 6.2.6 Common derivative method- 6.2.7 Integrating Factor- 6.2.8 Bernoulli equation- 6.3 Second Order Differential Equations- 6.3.1 Homogeneous second order ODE with constant coefficients- 6.3.2 Finding Solutions to 2nd order linear ODEs- 6.3.3 Non-homogeneous linear ODEs- 6.3.4 Applications of second order ODE
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Applications of second order ODEs
7.1 The driven simple harmonic oscillator
7.2 Unforced Oscillations
Limits and the evaluation of Indeterminate Forms
Series
9.1 Arithmetic Series
9.2 Geometric Series
9.3 Arithmetico-Geometric Series
9.4 A last example of a series
9.5 Convergent and Divergent Series
SummaryA. The Greek Alphabet
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A. The Greek Alphabet
Lists Greek letters with Greek symbols and their English equivalent.
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1 Introduction
Overview of the course and its relevance to Physics and Electronic Engineering.
Encouragement to practice through problem-solving.
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2 Functions
2.1 Definition of a Function
A function relates inputs (arguments) to outputs (values).
Domain, Codomain, and Range defined.
Example given: f(x) = x².
Additional notes on the properties of functions, including even/odd functions.
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2.2 Logarithms
Brief overview of logarithms and their definitions.
Important logarithmic identities and properties.
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2.2 Logarithms (cont'd)
Logarithmic rules: Product Rule, Quotient Rule, Power Rule, etc.
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2.3 Trigonometric Functions
Overview of basic trigonometric identities and properties.
Introduction to polynomial expansions of trigonometric functions.
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2.3 Trigonometric Functions (cont'd)
More identities, including reciprocals and angle sum identities.
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2.4 Power Series
Definition and importance of power series in mathematics.
Convergence of power series discussed.
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2.4.1 Taylor Series
Overview of Taylor Series expansion, how to derive coefficients, and examples.
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2.4.2 Approximation Errors in Taylor Series
Importance of understanding and estimating errors in Taylor approximations.
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3 Differential calculus
Overview of differentiable functions and the derivative's definition.
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3.1 Continuity of a function
Conditions for continuity at a point.
Examples of discontinuous functions.
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3.2 Differentiability and Differentiation
Definition of derivatives using limits and examples.
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3.3 Rules of Differentiation
Description of derivative rules, with examples on product, quotient, and chain rules.
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3.3 Rules of Differentiation (cont'd)
Additional rules, including logarithmic differentiation and Leibniz's Theorem.
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3.4 Special Points of a Function
Stationary points defined and different classifications of stationary points.
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3.4.1 Singular Points
Explanation of singularities in functions and implications.
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3.5 Inverse Functions
Definition and properties of inverse functions, alongside examples.
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3.5.1 Inverse Trigonometric Functions
Discusses the inverse trigonometric functions and their principal values.
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3.5.2 Inverse Function Rule
Derivation of derivative rules for inverse functions with examples.
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3.6 Partial Differentiation
Introduction to functions of multiple variables and their partial derivatives.
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4 A note on coordinate systems
Introduction to various coordinate systems and their applications.
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4.1 Circular Polar Coordinates
Definition and coordinate transformations for circular coordinates.
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4.2 Cylindrical Polars
Illustration and transformation of cylindrical coordinate systems.
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4.3 Spherical Polars
Explanation and utility of spherical coordinates in three-dimensional systems.
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5 Integration
Importance of integration as the inverse of differentiation illustrated through examples.
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5.1 Integration From First Principles
Discusses the fundamental concepts of integration and the formation of definite integrals.
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5.2 Integration of Arbitrary Continuous Function
The differences of positive and negative definite integrals are discussed, with examples.
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5.3 Integration as the inverse of Differentiation
The relationship between integration and differentiation is reinforced with typical problems.
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5.4 Integration By Inspection
Direct methods to integrate familiar forms and examples of their application.
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5.5 Integration by Substitution
Important techniques and examples for integrating functions by substitution.
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5.5.1 Indefinite Integrals
The significance of indefinite integrals via substitution methods.
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5.5.2 Solving Differential Equations with Integration
How integration plays a crucial role in solving differential equations, with practical examples.
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5.5.3 Definite Integrals
Techniques and principles of solving definite integrals, including boundary conditions.
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5.6 Integration by Parts
Justification and examples showing how integration by parts simplifies complex integrals.
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5.7 Differentiation of an Integral Containing a Parameter
Exploration of derivatives of integrals containing parameters provides a deeper understanding.
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5.8 An Aside on Hyperbolic Functions
Definitions and properties of hyperbolic functions, alongside their calculus-related identities.
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5.8.1 Identities of Hyperbolic Functions
Fundamental hyperbolic function equalities and their proof through manipulation.
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5.8.2 Inverses of Hyperbolic Functions
Explanation of the inverse functions of hyperbolic functions, along with properties and examples.
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5.8.3 Calculus of Hyperbolic Functions
Derivatives of hyperbolic functions are compared to trigonometric derivatives.
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5.9 Hyperbolic Functions In Integration by Substitution
The application of hyperbolic functions in substitution methods for integrals.
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6 Differential Equations
The definition and applications of ordinary differential equations (ODEs) are reciprocally connected to applied mathematics.
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6.1 Solving ODEs
Basic principles of ODEs and general solving methods illustrated through numerous examples.
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6.2 First Order ODE
Different types of first-order ODEs and categorizing methods of solving them appropriately.
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6.2.1 Separable First Order ODEs
The separation of variables method for solving first-order ODEs is discussed, substantiated with examples and applications.
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6.2.2 Almost Separable First Order ODEs
Tackling ODEs that aren't readily separable but can be transformed into a separable form.
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6.2.3 Homogeneous First Order ODEs
Introduction to homogeneous first-order ODEs, enabling reduction methods for solutions.
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6.2.4 Homogeneous Apart from a Constant
Exploring ODEs with constants can be turned into an equivalent homogeneous form allowing standard approaches to apply.
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6.2.5 Looks Homogeneous but Actually Separable with a Change of Variable
Insights into reviewing seemingly homogeneous equations closer to separable equations through transformation.
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6.2.6 Common Derivative Method
Examining simple first-order ODEs using common derivative methods, guiding stepwise analysis through to solutions.
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6.2.7 Integrating Factor
Detailed explanation of integrating factors and their critical application in solving linear first-order ODEs.
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6.2.8 Bernoulli Equation
Elucidation of the Bernoulli differential equations and their classifications with methods for finding solutions.
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7 Second Order Differential Equations
Overview of second-order differential equations and common forms; introduces initial conditions and expected general solutions.
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7.1 Homogeneous Second Order ODE with Constant Coefficients
Explains how to solve using characteristic equations and general solutions.
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7.2 Finding Solutions to 2nd Order Linear ODEs
Fundamental trial solutions and their derivations based on root conditions and unique coefficients defined from the polynomials.
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7.2.1 Real Roots
Detailed analysis of solving for second-order ODEs through real roots and examples.
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7.2.2 Complex Roots
Understanding how complex roots reflect solutions and applicability of Euler’s formula in visualizing results.
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7.2.3 Equal Roots and the Method of Reduction of Order
Special methods to deal with equations containing repeated roots through reduction techniques to derive more comprehensive solutions.
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7.3 Non-Homogeneous Linear ODEs
Procedures for finding complementary functions and establishing methods for particular integrals within the non-homogeneous framework.
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7.3.1 The Method of Undetermined Coefficients
Explanation of this calculating method to retrieve particular solutions through guesswork and confirming through substitution.
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7.3.2 Particular Solution When Non-Homogeneous Term is a Sum
A survey on how to separately deal with individual sums of functions and infer a cohesive overall solution.
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7.3.3 Use of Complex Exponentials in Solutions
Elaborates solutions utilizing complex exponentials to streamline resolving non-homogeneous equations efficiently.
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8 Applications of Second Order ODE
Discusses practical applications in physical systems and oscillators; includes derivations of general solution structures.
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8.1 The Driven Simple Harmonic Oscillator
Explanation of different forces acting on a mass-spring system and factors leading to distinct forms of motion.
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8.1.1 Unforced Oscillations
Comparison of various damping conditions with functional descriptions of their behaviors over time.
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9 Limits and the Evaluation of Indeterminate Forms
Introduction to limits, including definitions and various rules through examples.
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9.1 L'Hôpital's Rule
Application of L'Hôpital's Rule in simplifying and solving limits leading to clearer comprehension of indeterminate forms.
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10 Series
Fundamental definitions and descriptions of series, including conditions for convergence and divergence.
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10.1 Arithmetic Series
Definition and formula derivations for summing arithmetic series.
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10.2 Geometric Series
Defining geometric series and methods for summing both finite and infinite series.
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10.3 Arithmetico-Geometric Series
Explains the combination of arithmetic and geometric series characteristics, leading to particular summing strategies.
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10.4 Last Example of a Series
Exploration and proof of summing powers using relationships to integrate known equations.
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10.5 Convergent and Divergent Series
Clarifies definitions, tests for convergence, and the implications of divergent series.
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10.5.1 Testing a Series for Convergence: The Preliminary Test
Initial checks for convergence, establishing conditions under which series diverge.
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10.5.2 Tests for Convergence of Series of Positive Terms
Specific tests to discern convergences, such as Comparison Test and Integral Test.
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10.5.3 Alternating Series
Introduces tests specific to alternating series and methods for determining their behaviour.
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11 Summary
Review and advising the reader on the best practices for mastering calculus through continuous application and practice.