Module: Calculus of Several Variables & Fourier Analysis

Latest Module Specifications

Current Academic Year 2025 - 2026

Module Title	Calculus of Several Variables & Fourier Analysis
Module Code	MTH1047 (ITS: MS244)
Faculty	Mathematical Sciences	School	Science & Health
NFQ level	8	Credit Rating	7.5

Description

This module aims to ensure that students will have an understanding of the theory of the Calculus of Several Variables suitable for their studies in Physics. Where appropriate, theorems will be explained using arguments based on Physics rather than on formal mathematical proofs. Furthermore, it aims to develop students abilities to perform the calculations that arise in applications, especially in applications to Physics. Infinite dimensional vector spaces, in teh context of Fourier Series, will be considered also. Students will attend lectures on the course material and will work, independently, to solve problems on topics related to the course material. The students will have an opportunity to review their solutions, with guidance, at weekly tutorials.

Learning Outcomes

1. Reformulate concepts from physics in the language of vector calculus.
2. Perform the calculations that arise when the calculus of several variables is used to solve problems.
3. Demonstrate an understanding of concepts by use of examples or counterexamples.
4. State and apply selected definitions and theorems.
5. Calculate trigonometric Fourier Series of elementary functions defined on finite intervals and sketch the periodic extensions of such functions.

*Type*	*Hours*	*Description*
Workload	Full time hours per semester
Lecture	42	Students will attend lectures where new material will be presented and explained. Also attention will be drawn to various supporting material and tutorials as the course progresses.
Tutorial	18	Students will show their solutions to homework questions and will receive help with and feed-back on these solutions.
Independent Study	129.5	Corresponding to each lecture students will devote approximately 1.5 additional hour of independent study to the material discussed in that lecture or to work on support material when attention is drawn to such in lectures. Before each tutorial students will devote approximately three to solving homework problems which are to be discussed in that tutorial.
Total Workload: 189.5

Section Breakdown
CRN	20790	Part of Term	Semester 2
Coursework	0%	Examination Weight	0%
Grade Scale	40PASS	Pass Both Elements	Y
Resit Category	RC1	Best Mark	N
Module Co-ordinator	Niamh O'Sullivan	Module Teacher

Type	Description	% of total	Assessment Date
Assessment Breakdown
Loop Quiz	n/a	10%	n/a
Assignment	n/a	10%	n/a
In Class Test	n/a	20%	n/a
Assignment	n/a	10%	n/a
Formal Examination	n/a	50%	End-of-Semester

Reassessment Requirement Type
Resit arrangements are explained by the following categories; RC1: A resit is available for both^* components of the module. RC2: No resit is available for a 100% coursework module. RC3: No resit is available for the coursework component where there is a coursework and summative examination element. ^* ‘Both’ is used in the context of the module having a coursework/summative examination split; where the module is 100% coursework, there will also be a resit of the assessment

Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml

Indicative Content and Learning Activities

VECTORS IN 3-SPACE
Vectors as directed line segments, their addition and scalar multiplication. Frames of reference and coordinates. The inner product, cross product and their applications.

FUNCTIONS OF ONE OR MORE VARIABLES
Parametrized curves. Level sets and their parametrizations.

DIFFERENTIAL CALCULUS
Limits, continuity, partial differentiation. The chain rule. The gradient, divergence, curl and their physical interpretations. Max/Min problems and Lagrange multipliers. Taylor's formula.

INTEGRAL CALCULUS
Line integrals, multiple integrals, surface integrals. The integral theorems. Change of variable formula for multiple integrals.

FOURIER SERIES
Function spaces. Orthogonal projections onto finite dimensional spaces. Calculation of Trigonometric Fourier Series, Bessel's inequality, Parseval's identity. Fourier Transforms.

Indicative Reading List

Books:
None

Articles:
None

Other Resources

None