###### Learning Outcomes:

Students must be able to calculate the quantities inherent in the course: tangent vector, tangent line, tangent space, unit-speed reparametrisations etc.. They must be able to state the theorems proved in the course and reproduce some elements of proof. They must demonstrate an understanding of the concepts involved.

More generally students should be able to do the following:

WRITE MATHEMATICS: Students should be able to recognise, read and correctly use standard mathematical symbols and notation, to correctly write a mathematical statement and to recognise when such a statement is not correctly written.

QUESTION: Students should be able to ask pertinent questions themselves, to decide which questions are most relevant, which questions are answerable, which questions they should start with, etc.

UNDERSTAND: Students must be able to understand the reasoning behind any methods or procedures they use and be able to demonstrate that understanding.

PRODUCE EXAMPLES: Students must be able to produce examples themselves, to illustrate a definition, to show a method, to test boundaries of an idea.