M4: Geometry (2016-2017)
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15 lectures
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.
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.
See the examinable syllabus
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. $\mathbf{v} = \omega \land \mathbf{r}$. [1.5]
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. [3.5]
1) J. Roe, Elementary Geometry (Oxford Science Publications, 1992), Chapters 1, 2.2, 3.4, 4, 5.3, 7.1--7.3, 8.1--8.3, 12.1.
2) M. Lunn, A First Course In Mechanics (Oxford Science Publications, 1991) Chapter 4