# ACM20150 Vector Integral & Differential Calculus

## Academic Year 2022/2023

This module introduces the fundamental concepts and methods in the differentiation and integration
of vector-valued functions and also provides an introduction to the Calculus of Variations.

[Fundamentals] Vectors and scalars, the dot and cross products, the geometry of lines and planes,
[Curves in three-dimensional space] Diferentiation of curves, the tangent vector, the Frenet-
Serret formulas, key examples of Frenet-Serret systems to include two-dimensional curves, and
the helix, [Taylor's theorem in one and several variables] Conditions for the convergence of Taylor series, practical computations, extension to Taylor's theorem in several variables, the connection with the differential of a multi-variable function, [Partial derivatives and vector fields] Introduction to partial derivatives, scalar and (Cartesian) vector fields, the operators div, grad, and curl in the Cartesian framework, applications of vector differentiation in electromagnetism and fluid mechanics, [Mutli-variate integration] Area and volume as integrals, integrals of vector and scalar fields, Stokes's and Gauss's theorems (statement and proof), [Consequences of Stokes's and Gauss's theorems] Green's theorems, the connection between vector fields that are derivable from a potential and irrotational vector fields, [Curvilinear coordinate systems] Basic concepts, the metric tensor, scale factors, div, grad, and curl in a general orthogonal curvilinear system, special curvilinear systems including spherical and cylindrical polar coordinates

Further topics may include: Introduction to differential forms, exact and inexact differential forms,
[Advanced integration] Integrating the Gaussian function using polar coordinates, the gamma func-
tion, the volume of a four-ball by appropriate coordinate parameterization, the volume of a ball in an
arbitrary (finite) number of dimensions using the gamma function,
[Applications in general relativity] Lengths and volumes in curved spacetime.
[Fluid mechanical application] Incompressible flow over a wavy boundary, [Calculus of variations] Constrained variations.

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Curricular information is subject to change

Learning Outcomes:

On completion of this module students should be able to

1. Write down parametric equations for lines and planes, and perform standard calculations based
on these equations (e.g. points/lines of intersection, condition for lines to be skew);
2. Compute the Frenet-Serret vectors for an arbitrary differentiable curve;
3. Compute the series expansion of important functions using Taylor's theorem;
4. Differentiate scalar and vector fields expressed in a Cartesian framework;
5. Perform operations involving div, grad, and curl;
6. Perform line, surface, and volume integrals. The geometric objects involved in the integrals
may be lines, arbitrary curves, simple surfaces, and simple volumes, e.g. cubes, spheres,
cylinders, and pyramids;
7. State precisely and prove Gauss's and Stokes's theorems;
8. Derive corollaries of these theorems, including Green's theorems and the necessary and sufficient condition for a vector fueld to be derivable from a potential;
9. Compute the scale factors for arbitrary orthogonal curvilinear coordinate systems;
10. Apply the formulas for div, grad, and curl in arbitrary orthogonal curvilinear coordinate systems;

Student Effort Hours:
Student Effort Type Hours
Autonomous Student Learning

40

Lectures

36

Tutorial

24

Total

100

Approaches to Teaching and Learning:
Lectures, tutorials, enquiry, and problem-based learning
Requirements, Exclusions and Recommendations
Learning Requirements:

A good knowledge of calculus

Learning Recommendations:

Students should have followed:
1. MATH10300 - Calculus in the Mathematical Sciences
or
2. MATH10330 - Calculus in the Physical Sciences

Module Requisites and Incompatibles
Equivalents:
Vector Calculus (MAPH20150)

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Continuous Assessment: Homework assignments and midterm test Varies over the Trimester n/a Standard conversion grade scale 40% No

30

Examination: Final exam 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

70

Carry forward of passed components
No

Resit In Terminal Exam
Autumn Yes - 2 Hour
Feedback Strategy/Strategies

• Group/class feedback, post-assessment

How will my Feedback be Delivered?

Not yet recorded.

Name Role
Mr Chris Devitt Tutor
Mr Jake Williams Tutor
Timetabling information is displayed only for guidance purposes, relates to the current Academic Year only and is subject to change.

Spring

Lecture Offering 1 Week(s) - 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33 Fri 13:00 - 13:50
Lecture Offering 1 Week(s) - 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33 Thurs 14:00 - 14:50
Lecture Offering 1 Week(s) - 20, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33 Tues 14:00 - 14:50
Tutorial Offering 1 Week(s) - 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33 Tues 11:00 - 11:50
Tutorial Offering 2 Week(s) - 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33 Tues 10:00 - 10:50
Spring