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?


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


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.)

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.