My view of differential calculus

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.