## MATH1051Calculus II

### Module Overview

Core for all Programmes in the School of Mathematics.

### Module Details

Title: Calculus II
Code: MATH1051
Year: 1
Semester: 2

CATS points: 15 ECTS points: 7.5
Co-ordinator(s): Dr James Anderson

### Pre-requisites and / or co-requisites

ModuleCodeYear
Linear Algebra IMATH10481
Calculus IMATH10501
Differential EquationsMATH10521

### Aims and objectives

• Evaluate and interpret derivatives of functions of two or more variables;
• Determine differentiability and calculate derivatives of functions of a complex variable;
• Set up and evaluate double and triple integrals over simple regions.

### Syllabus

I. Differentiable calculus of functions of several variables:

Sketching functions of two variables.

Partial derivatives, tangent plane, maximum and minimum points.

Change of variables and the chain rule, Jacobian.

Second and higher partial derivatives, change of variables for second derivatives.

Critical points and their classification.

II. Differentiable calculus of functions of a complex variable:

Algebra of complex numbers, including Cartesian and polar representation.

Euler’s formula and applications to trigonometry; De Moivre’s theorem and roots of unity.

Complex differentiation, Cauchy-Riemann equations, holomorphic functions.

III. Integral calculus of functions of two or three variables:

Double and repeated integrals, including change of order in a repeated integral.

General change of variables in a double integral.

Polar coordinates, cylindrical polar coordinates, spherical polar coordinates.

Triple integrals in various coordinates.

Line integrals.

### Study time allocation

Contact hours: 4
Private study hours: 4
Total study time: 8 hours

### Teaching and learning methods

Lectures, weekly small group tutorials, private study

For parts I and III of the syllabus, a good source is ADAMS R A, Calculus - A complete course. (Addison-Wesley)

For part II of the syllabus, a good source is BROWN JW & CHURCHILL RV Complex variables and Applications (McGraw-Hill)

### Assessment methods

• 70% written examination;
• PURPOSE OF ASSESSMENT: To allow students to demonstrate that they have understood the material and can carry out basic calculations correctly
• FEEDBACK: Students will be emailed general comments on the performance of the class as a whole within a few weeks of the examination along with examination solutions
• 10% class test;
• PURPOSE OF ASSESSMENT: To provide students with a snapshot of their current understanding of the material being tested
• FEEDBACK: Students’ work will be marked and returned within a week of the test. Solutions will be posted on blackboard
• 20% Coursework;
• PURPOSE OF ASSESSMENT: To provide regular practice, supported by feedback on the material and techniques taught in the module
• FEEDBACK: Student’s work will be marked and returned within a week of hard in. Model solutions will also be posted on Blackboard
• Referral Assessment: Written Examination.