Mathematical Methods for Physical Scientists 1a

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• Construct a Taylor series of a function and understand its relevance to local behaviour
• Solve simple polynomial equations
• Integrate various simple functions of one variable
• Algebraically manipulate 3 by 3 matrices, and solve eigenvalue problems in 2 dimensions;
• Understand the concept of a limit and be able to determine its value if it exists
• Calculate the scalar and vector product of two vectors;
• Differentiate functions of several variables and manipulate then when changing variables
• Sketch and manipulate exponential, trigonometric and hyperbolic functions
• Differentiate functions of one variable and use this to classify critical points

• calculate the scalar and vector product of two vectors;
• algebraically manipulate 3 by 3 matrices, and solve eigenvalue problems in 2 dimensions;
• solve simple polynomial equations, including complex solutions
• sketch and manipulate exponential, trigonometric and hyperbolic functions;
• differentiate functions of one variable and use this to classify critical points;
• understand the concept of a limit and be able to determine its value if it exists;
• construct a Taylor series of a function and understand its relevance to local behaviour;
• understand the nature of simple complex valued functions;
• differentiate functions and manipulate them when changing variables;
• integrate various functions of one variable.
• Basic vector algebra, cross and dot product, geometrical and physical applications.

Matrices and determinants, inverse of a matrix, using matrices to solve simultaneous equations. Eigenvalue problems.

Solving quadratic equations, factorising higher order polynomials.

Complex numbers, powers of i, Argand diagrams, modulus and argument of a complex number, complex conjugates. The algebra of complex numbers: addition, subtraction, multiplication and

division in both Cartesian and polar form. Specifying a function, its domain and range. Composition of functions. Graphs of functions. One-toone functions, inverse functions and their graphs. Even and odd functions, periodic functions, trigonometric functions, inverse trigonometric functions. Informal definition of a limit, rules for evaluating limits, infinite limits. Rules for differentiation, higher derivatives, critical points and applications to graph sketching. Exponential and natural logarithm functions, power functions, hyperbolic functions, inverse hyperbolic functions and their derivatives. Derivatives of vectors L'Hôpital's Rule, Taylor series expansions and remainder terms.

Complex exponentials and trigonometric functions. De Moivre's Theorem, calculating powers and roots, and solving equations.

Integration, the Fundamental Theorem of Calculus, indefinite integrals, methods of integration, partial fractions, integration by parts.

Lectures, small group tutorials, private study. The method of delivery in lectures will be “chalk and talk”, however the students will be provided with printed skeletal notes, highlighting all the key results and saving them from excessive note taking. Hard copies of the skeletal notes and all the assignments will be provided, and the material will also be provided in the module Blackboard site.

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.