Theoretical physics is dominated by partial differential equations
such as the Euler equation, which you have probably seen written out
in Cartesian coordinates. But what form does it take in spherical
polar coordinates? Or in an arbitrary coordinate system? What if space
(or spacetime) is curved, as general relativity tells us it is?
A fundamental idea of modern physics is that all its laws should be
geometric in nature, that is they should be relations between
geometric quantities such as a velocity vector field, independent of
the coordinates used to describe this object. These objects could live
in the 3-dimensional space of our experience and of Newtonian physics,
or they could live in the the 4-dimensional spacetime of relativistic
physics.
A more abstract example is the state of a gas in thermodynamical
equilibrium. Its state is fixed by any three of the following
properties: its volume, pressure, temperature, internal energy,
entropy, chemical potential. All remaining properties can then be
treated as functions of the selected three. A lot of the mathematical
difficulty in elementary thermodynamics can be avoided by treating the
space of all equilibrium states as a (3-dimensional, in this case)
manifold. (As you will learn, a manifold is, roughly speaking, a space
that is locally like
Rn.) Similarly, it is more useful to treat
3-dimensional space or 4-dimensional spacetime as manifolds, rather
than as vector spaces
R3 or
R4.
This course will teach you all the core mathematical concepts you need
for writing physical laws in geometric form first, and only then use
them to introduce a few selected areas of physics where a geometric
view is either essential, or really makes things easier.
Undergraduate calculus, in particular integration in several
variables. Undergraduate linear algebra, in particular abstract vector
spaces. Vector calculus would be useful but is not essential.