Module: Calculus of Several Variables & Fourier Analysis

Module Specifications.

Current Academic Year 2024 - 2025

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Date posted: September 2024

Module Title	Calculus of Several Variables & Fourier Analysis
Module Code	MS244 (ITS) / MTH1047 (Banner)
Faculty	Science & Health	School	Mathematical Sciences
Module Co-ordinator	Ronan Egan
Module Teachers	-
NFQ level	8	Credit Rating	7.5
Pre-requisite	Not Available
Co-requisite	Not Available
Compatibles	Not Available
Incompatibles	Not Available

None

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

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.

Assessment Breakdown
Continuous Assessment	50%	Examination Weight	50%

Type	Description	% of total	Assessment Date
Course Work 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

Reassessment Requirement Type
Resit arrangements are explained by the following categories: Resit category 1: A resit is available for both* components of the module. Resit category 2: No resit is available for a 100% continuous assessment module. Resit category 3: No resit is available for the continuous assessment component where there is a continuous assessment and examination element.
* ‘Both’ is used in the context of the module having a Continuous Assessment/Examination split; where the module is 100% continuous assessment, there will also be a resit of the assessment
This module is category 1

Indicative Reading List

Other Resources

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