ADVANCED CALCULUS AND CURVATURE
MAT3AC
2017
Credit points: 15
Subject outline
This subject unifies linear algebra and vector calculus from second year into the sort of calculus used in differential geometry and analysis. The central notion is that of manifolds, which is a generalization of curves and surfaces, and provides the mathematical context for understanding "space" in all of its manifestations. Topics include: the implicit and inverse function theorems, manifolds and tangent space, curvature and torsion for curves, intrinsic and extrinsic curvature for surfaces, and the Theorema Egregium of Gauss.
School: School Engineering&Mathematical Sciences
Credit points: 15
Subject Co-ordinator: Yuri Nikolayevsky
Available to Study Abroad Students: Yes
Subject year level: Year Level 3 - UG
Exchange Students: Yes
Subject particulars
Subject rules
Prerequisites: MAT2VCA
Co-requisites: N/A
Incompatible subjects: N/A
Equivalent subjects: N/A
Special conditions: N/A
Learning resources
Readings
| Resource Type | Title | Resource Requirement | Author and Year | Publisher |
|---|---|---|---|---|
| Readings | Advanced Calculus and Curvature | Prescribed | Y.Nikolayevsky, 2014 | LTU |
Graduate capabilities & intended learning outcomes
01. Construct manifolds explicitly and implicitly, as well as their tangent spaces
- Activities:
- These are modelled in lectures, with reinforcment in practice classes and marked feedback through assignments
- Related graduate capabilities and elements:
- Inquiry/ Research(Inquiry/ Research)
- Creative Problem-solving(Creative Problem-solving)
- Critical Thinking(Critical Thinking)
- Discipline-specific GCs(Discipline-specific GCs)
- Writing(Writing)
02. Use MAPLE to graph manifolds and investigate mappings
- Activities:
- Guided exercises in practice classes, with marked feedback through assignments
- Related graduate capabilities and elements:
- Inquiry/ Research(Inquiry/ Research)
- Quantitative Literacy/ Numeracy(Quantitative Literacy/ Numeracy)
- Critical Thinking(Critical Thinking)
- Creative Problem-solving(Creative Problem-solving)
- Discipline-specific GCs(Discipline-specific GCs)
03. Correctly invoke the inverse and implicit function theorems in mathematical arguments
- Activities:
- Demonstrated in lectures, with reinforcment in practice classes and marked feedback through assignments
- Related graduate capabilities and elements:
- Inquiry/ Research(Inquiry/ Research)
- Writing(Writing)
- Creative Problem-solving(Creative Problem-solving)
- Critical Thinking(Critical Thinking)
- Discipline-specific GCs(Discipline-specific GCs)
04. Calculate the curvature of curves and surfaces
- Activities:
- Demonstrated in lectures, with reinforcment in practice classes and marked feedback through assignments
- Related graduate capabilities and elements:
- Writing(Writing)
- Discipline-specific GCs(Discipline-specific GCs)
- Quantitative Literacy/ Numeracy(Quantitative Literacy/ Numeracy)
- Creative Problem-solving(Creative Problem-solving)
- Inquiry/ Research(Inquiry/ Research)
- Critical Thinking(Critical Thinking)
05. Communicate your understanding (of manifolds) using both words and mathematical notation in a precise and succinct manner
- Activities:
- Written communication is demonstrated through lectures, with guidance in practice class and with marked feedback in assignments. Information and guidance is also provided to aid the writing of the essay.
- Related graduate capabilities and elements:
- Writing(Writing)
- Critical Thinking(Critical Thinking)
- Discipline-specific GCs(Discipline-specific GCs)
- Inquiry/ Research(Inquiry/ Research)
- Creative Problem-solving(Creative Problem-solving)
Melbourne, 2017, Semester 2, Day
Overview
Online enrolment: Yes
Maximum enrolment size: N/A
Enrolment information:
Subject Instance Co-ordinator: Yuri Nikolayevsky
Class requirements
PracticalWeek: 31 - 43
One 1.0 hours practical per week on weekdays during the day from week 31 to week 43 and delivered via face-to-face.
LectureWeek: 31 - 43
Two 1.0 hours lecture per week on weekdays during the day from week 31 to week 43 and delivered via face-to-face.
Assessments
| Assessment element | Comments | % | ILO* |
|---|---|---|---|
| Four written mathematics assignments | 20 | 01, 02, 03, 04, 05 | |
| One 1000-1500 word essay | 10 | 05 | |
| One 2-hour written exam | 70 | 01, 03, 04, 05 |