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Although I claim to do geometry, as those of you who learned
calculus may know, much geometry can be understood with the help
of calculus. Thus we shall use some (okay, a lot) of calculus to
help us prove some interesting things about geometry. At least,
that's the plan. But DON'T FEAR!!!! I will hopefully give you a
good idea of what you need to know about calculus. So if you
don't know a thing about it, now you can get the gist.
So before we start with the math, perhaps we should motivate. One
view (maybe Newton's, for one) of how things work is there are
these forces always working and everything happens due to these
forces which are always working. For instance, if we have a piece
of cork floating in a river, there are currents acting on it at
all time and that causes the cork to move. The important point is
that the forces are ALWAYS working on it. In contrast, we could
think of the second hand on a clock that clicks to the next tic
mark every second. This isn't changing continuously like the
cork, but a force is acting on it at discrete instances, that is,
once every second. Calculus is not designed for this type of
situation, but for the first one.
So based on the philosophy that there are forces acting on our
"objects" (by this I mean whatever we want to consider: the cork's
spatial location, it's velocity, it's density, or anything else),
we need something that measures how they change. So if we have a
cork in a river, it will be moved by the currents and we want some
way of understanding how the current causes the position of the
cork to change. This is something called a vector field. At
every point in the river, it tells where the cork is going to go.
Well, not quite, since after it moves a little bit, it is at
another point and thus moves according to another vector (all a
vector is is an arrow that tells me which direction to move and
how fast). This is the magic of calculus. It lets us deal with
what seems like a mess. Essentially you move in a straight line
in the direction the vector says, but only for a really small
amount of time, and then you are at a new point and move in a straight line
according to the vector there for a small amount of time, and so
on. This is called the flow of a vector field (because it is the
way the water flows in the river).
This is essentially all a differential equation is. It associates
a vector (which says where to go) to each point in your space.

** Next:** What is a manifold?
** Up:** An Overview of My
** Previous:** Introduction
David Glickenstein
2003-12-07