Date: 28 April 2017
- For functions of one variable, application of calculus.
- Have first derivative and second derivative test.
- In higher dimension:
- to define first derivative: linear map (actually functional in this case).
- second derivative : also captureb by a linear map - but a square symmetric matrix.
- we can also have interesting constraints.
- even more generally, we have calculus of variations.
- Study linear maps on vector spaces, because:
- many maps are linear
- many functions are quadratic - these also can be expressed in linear algebra
- smooth functions are locally approximately linear, with the second term quadratic.
- By study, we typically mean properties or structures that do not change when we chang co-ordinates.
- for a vector space, there is just one, the dimension.
- for linear map, we consider different classes.
- domain and co-domain equal (and change co-ordinates together).
- co-domain real numbers (functionals).
- more structure on vector spaces to be preserved - e.g. lengths and angles.
- special classes of maps, e.g. symmetric matrices.
- One linear map can be described completely:
- Sylvester’s law of inertia
- Jordan canonical form
Local to global
- Differential calculus lets us study local properties of functions.
- To deduce from this global behaviour, we integrate and solve differential equations.