This will turn into a standard equation of the type dv/dt=kv+ noise after a change of variable. For some general methods for solving SDEs, I hope the following link will be of much help -
http://math.berkeley.edu/~evans/SDE.course.pdf
Have you come across Korner's ' A Companion to analysis'? I find it a good second course, even after having studied Rudin's & Apostol's books.
Separate textbooks on complex analysis & on differential geometry would help , as these have much special attributes. An overview of books...
It won't be simply the sum of the two. It will consist of alternate pieces of the graphs of the two. We must consider the time intervals in which x>0 & x<0 separately.
Have you tried 'Differential Equations' by Simmons? It's a good book & contains many problems in physics.
For a course in PDEs, you must be familiar with calculus of several variables & a little functional analysis.
The curvature of potential energy is definitely related to distribution function, but the dependence is weird.I can't give any physical meaning to curvature which might suit the dependence.
I can't think of a definition of the lie derivative with respect to a covector off my head.
However, we may talk about the lie derivative with respect to a totally contravariant tensor.We could define it as the tensor product of component-wise lie derivatives. Such a quantity could be another...
Think of a sigma algebra as 'containing information'.Since G is the trivial sigma algebra, it contains no intrinsic information & doesn't affect the expectation.
I must admit that this terminology is vague & nearly metaphorical. It's perfectly fine if you stash this terminology if it...
Let a surface be X=(x(u,v),y(u,v),z(u,v)) . The element of area will be the cross product of Xu & Xv ( Xu is the partial derivative with respect to u &c.).Integrate this within the limits.
(a) Let M be the upper bound for f & consider the sets
O_n={x in D |2M/(n+1)<=Of(x)<=2M/n} n=1,2,..
Do you see why each O_n must be closed & their union is D?
(b)If D is a null set,f would be riemann integrable(& hence measurable) as it is bounded .
We must check whether all the limits exist in the first place.Although the general question is difficult to answer,I have a counterexample for the second ( where the second limit doesn't exist).
No, a boundary point may not be an accumulation point.Since an isolated point has a neighbourhood containing no other points of the set, it's not an interior point. The boundary is, by definition , A\intA & hence an isolated point is regarded as a boundary point.