ma01c09rec4-9 - Ma1c Analytic Recitation 4/9/09 1 Topology...

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Unformatted text preview: Ma1c Analytic Recitation 4/9/09 1 Topology Most of your homework questions last week were all about proving things about open sets. Really, you were studying R n with its standard topology . A topological space is a pair ( X, T ) where X is a set and T is a collection of subsets of X , which are called the open sets of X . The open sets must satisfy the following: • X and ∅ are in T . • A finite interesection of sets in T is in T (a finite intersection of open sets is open). • A arbitrary union of open sets is open A example of a topological space is ( R n , T ), where T is all unions of open balls. This is the standard topology that you are used to and you used in all of your homework problems. There are other topologies on R n , however. Consider the space ( R , T ), where T is all complements of finite sets. This is called (I think) the finite complement topology. An example of a set in T is R \{ , 1 , 2 , 3 } . This topology has the interesting property that every sequence of distinct values converges to every point. Why is this so? Given any sequence and point and open set about the point, there are only a finite number of points outside the open set. Therefore, there is some natural number past which the sequence must lie entirely in the open set. The general definition of continuity is in the language of topology, also. It says that a function is continuous if the preimage of any open set is open. You can check that this is equivalent to the usual definition. 2 Homework Note On problem 8.14.8, it gives you a hint to use problem 7(d). You may use the result of exercise 7(d) without proof (but you should say “here I am using the result of 7(d)”). 3 Directional Derivatives In all of the following, we will consider a function f : R n → R , i.e. a scalar field. We restrict the range to R because this case is simpler and there are more tricks you case use to get a handle on what’s going on. If the domain of the function is R , then you know how to differentiate the function, but what does that mean in higher dimensions? To differentiate, you find, intuitively, the instantaneous change in the function when you move in the domain. In R , you don’t have to think about which direction you are moving, but in R n , there are lots of directions! The easiest way to generalize the notion of a one-dimensional derivative is to use a directional derivative: f ( a ; y ) = lim h → f ( a + h y )- f ( a ) h Where y is a unit vector! When the limit exists. Notice this is really just the one-dimensional derivative of the one-dimensional function g : R → R defined by g ( t ) = f ( a + t y ). The most common directions that you want to know about are the basis directions. Thus, we call f ( a , e k ) the partial derivative of f with respect to x k . It is also denoted by ∂f ∂x k (and a few other things you can find on page 254). You can, of course, take(and a few other things you can find on page 254)....
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This note was uploaded on 07/19/2010 for the course MA 1C taught by Professor Ramakrishnan during the Spring '08 term at Caltech.

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ma01c09rec4-9 - Ma1c Analytic Recitation 4/9/09 1 Topology...

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