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 |
---|---|

Lectures | 36 |

Tutorial | 24 |

Autonomous Student Learning | 40 |

Total | 100 |

Approaches to Teaching and Learning:

Lectures, tutorials, enquiry, and problem-based learning

Lectures, tutorials, enquiry, and problem-based learning

Requirements, Exclusions and Recommendations

A good knowledge of calculus

Students should have followed:

1. MATH10300 - Calculus in the Mathematical Sciences

or

2. MATH10330 - Calculus in the Physical Sciences

Module Requisites and Incompatibles

Vector Calculus (MAPH20150)

Assessment Strategy

Description | Timing | Component Scale | % of Final Grade | ||
---|---|---|---|---|---|

Not yet recorded. |

Carry forward of passed components

No

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 |