Advanced Mathematics Course Details

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ADVANCED MATHEMATICS

MONASH UNIVERSITY FOUNDATION YEAR

1. INTRODUCTION

A student taking this course must also be concurrently enrolled in (or previously studied) MUFY Mathematics Part A as many of the topics in MUFY Advanced Mathematics require an understanding of the concepts in MUFY Mathematics Part A.

2. COURSE OBJECTIVES

Advanced Mathematics is designed to prepare students who wish to take tertiary courses with a high mathematical content, or which use a considerable amount of mathematical reasoning. In Part A, students study matrices, complex numbers, vectors, trigonometric functions and differentiation techniques. In Part B the topics covered are integration techniques and applications of definite integrals, differential equations and kinematics.

3.COURSE CONTENT

Semester A:
1. Matrices & Linear Algebra

The concept of a matrix; matrix algebra, including addition, subtraction, and multiplication of matrices, and multiplication of a matrix by a scalar. The conditions necessary for the sum or product of matrices to exist.

The unit matrix, I; the meaning of the inverse, A-1, of a matrix A; the fact that AA-1 = A-1A = I.

Determinants; the determinant of a 2 x 2 matrix; the inverse of a 2 x 2 matrix. The use of matrices to solve systems of two equations in two unknowns.

Singular matrices; the fact that, if a matrix is singular, the equations it represents must be either dependent or inconsistent.

2. Complex Numbers

Algebraic form [pic] where [pic] and where x, y are real numbers. The terminology complex plane. The real part, and imaginary part, of z defined. Addition and subtraction defined algebraically. Multiplication based on the definition[pic].

The complex conjugate [pic] of [pic]. Property [pic]. Modulus of z. Division by complex numbers.

Polar form for non-zero z; arguments [pic] of z. Principal Argument, Arg z: [pic]. Subsets of the complex plane such as [pic] or [pic] sketched; known as Argand diagrams.

De Moivre’s Theorem for integer index introduced and used to find (two) square roots, (three) cube roots and (four) fourth roots for non-zero complex numbers.

Awareness of the Fundamental Theorem of Algebra., which allows complex solutions to be found for any polynomial with real coefficients; and the fact that if any complex number z is a solution of a given polynomial, then [pic]is also a solution.. Use of the Factor Theorem to factor a real polynomial into first degree complex factors and to solve real polynomial equations for all complex number solutions.

3. Vectors

Vectors a (or [pic]) as abstract models of quantities such as spatial displacement, velocity or force. Vector algebra. The triangle (parallelogram) law of addition. Zero vector and subtraction defined. Scalar multiplication ta defined; algebraic properties.

Magnitude |a| of a vector a. Unit vectors. The canonical unit vectors i, j, k of cartesian 3-space.

Resolution of a vector in component form. Direction cosines of a non-zero vector a defined Examples including displacement, velocity, acceleration.

Scalar (dot) product [pic]defined. Algebraic properties. Definition of scalar resolute and vector resolute of a in the direction of non-zero vector b.

Applications in two and three dimensions, involving angles.

4.Circular Functions ( Trigonometric Functions)

Pythagorean identities:
[pic]

Sum and difference expansions:
[pic]
Deduction of double-angle expansions for [pic].
Solution of trigonometric equations.

The restricted one-to-one functions Sin, Cos, Tan and their inverses [pic]. Graphs. Simple modifications, e.g. [pic].

5. Introduction to Differential Calculus

The concept of limits, and...
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