ACM20150 Vector Integral & Differential Calculus

Academic Year 2023/2024

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




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
2. MATH10330 - Calculus in the Physical Sciences

Module Requisites and Incompatibles
Vector Calculus (MAPH20150)

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


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



Carry forward of passed components
Resit In Terminal Exam
Autumn Yes - 2 Hour
Please see Student Jargon Buster for more information about remediation types and timing. 
Feedback Strategy/Strategies

• Group/class feedback, post-assessment

How will my Feedback be Delivered?

Not yet recorded.

Name Role
Mr Chris Devitt Tutor