M4: Geometry - Material for the year 2019-2020

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2019-2020
Lecturer(s):
Dr Richard Earl
Course Term:
Michaelmas
Course Lecture Information:

15 lectures

Course Overview:

The course is an introduction to some elementary ideas in the geometry of euclidean space through vectors. One focus of the course is the use of co-ordinates and an appreciation of the invariance of geometry under an orthogonal change of variable. This leads into a deeper study of orthogonal matrices, of rotating frames, and into related co-ordinate systems.

Learning Outcomes:

Students will learn how to encode a geometric scenario into vector equations and meet the vector algebra needed to manipulate such equations. Students will meet the benefits of choosing sensible co-ordinate systems and appreciate what geometry is invariant of such choices.

Course Syllabus:
Course Synopsis:

Euclidean geometry in two and three dimensions approached by vectors and coordinates. Vector addition and scalar multiplication. The scalar product, equations of planes, lines and circles. [3]

The vector product in three dimensions. Use of $\mathbf{a}, \mathbf{b}, \mathbf{a} \land \mathbf{b}$ as a basis. $\mathbf{r} \land \mathbf{a} = \mathbf{b}$ represents a line. Scalar triple products and vector triple products, vector algebra. [2]

Conics (normal form only), focus and directrix. Showing the locus $Ax^2 + Bxy + Cy^2 = 1$ can be put in normal form via a rotation matrix. Orthogonal matrices. $2\times 2$ orthogonal matrices and the maps they represent. Orthonormal
bases in $\mathbb{R}^3$. Orthogonal change of variable; $A\mathbf{u} \cdot A\mathbf{v} = \mathbf{u \cdot v}$ and $A(\mathbf{u} \land \mathbf{v}) = \pm A\mathbf{u} \land A \mathbf{v}$. Statement that a real symmetric matrix can be orthogonally diagonalized. Simple examples identifying conics not in normal form. [3]

$3 \times 3$ orthogonal matrices; $SO(3)$ and rotations; conditions for being a reflection. Isometries of $\mathbb{R}^3$. [2]

Rotating frames in $2$ and $3$ dimensions. Angular velocity. $\mathbf{v} = \omega \land \mathbf{r}$. [1]

Parametrised surfaces, including spheres, cones. Examples of coordinate systems including parabolic, spherical and cylindrical polars. Calculating normal as $\mathbf{r}_u \land \mathbf{r}_v$. Surface area. [4]