# Different aspects of differential equations

I sometimes encountered the keywords such as 'geometric', 'analytical' aspects of differential equations, especially in general introduction. However I do not really understand what they are referring to. Could anyone clarify on those terminologies?

update:

For example, as mentioned in Tao's comment, what does exactly the geometric information of NS equation refer to?

## 1 Answer

The two aspects are really just ways of looking at differential equations. Take, for example, the differential equation of $y'=f(x,y)$.

• The analytical view of this equation is basically just finding the function $y$ for which $y'=f(x,y)$ holds. In a way, it looks at an operator on a certain subset of functions which maps the function $x\mapsto y(x)$ to the function $x\mapsto y'(x) - f(x, y(x))$. What the analytical view looks for is the "kernel" of this operator, that is the function(s) $y$ that map to the zero function.
• The geometric view, on the other hand, looks more at the graph of the function $y$. Because the function $x\mapsto y(x)$ has a graph, defined as $\{(x,y(x)| x\in D\}$ where $D$ is $y$-s domain. If you look at the tangent on the curve $(x,y(x))$, you see that it is $(1,y'(x))$, and because $y$ solves the differential equation, the tangent is in fact $(1, f(x,y(x))$. Now imagine the space $D\times \mathbb R$. In every point $(x,y)$ in the space, you have a vector $(1, f(x,y))$, so you have a mass of vectors (arrows) on a plane (a vector field). The idea of the "geometrical aspect" is that solving the equation $y'=f(x,y)$ is equivalent to finding such a curve that at every point on the curve, the vector field is tangent to the curve.

(DISCLAIMER: there is much, much more to the story than what is explained below. There are objects - distributions - , other than functions, that solve some differential equations, there are vector norms you must consider in the analytical view, the list is endles. I used the example above to give you an idea on the subject. My general advice is that you should not bother with "which view am I using now?" and try to learn as much as possible. Don't worry, a more general overview will build itself slowly, but surely, in your mind.)

• Thanks for your explanation. As mentioned in Tao's comment, what does exactly the geometric information of NS equation refer to? – newbie Feb 7 '14 at 14:44
• I don't know enough about these equations, sorry. – 5xum Feb 7 '14 at 14:49