Dear Professor Taylor,
I am having trouble apply open and connected to a
set. I'm looking back into the book and the lecture notes and I see how it
is applied to the path of a line integral but I don't see how it applies to
a set.
Can you explain these concepts again?
Thank you,
*****************************
"Connected" means that between any two points in the set there is a path lying completely in the set that connects those two points. Here are three connected regions and one disconnected one, crossed out.
Open means that every point inside the set is some nonzero distance away from points outside the set--think of the open unit interval {x:0<x<1}: even points really close to 0, or 1 are some nonzero distance from points outside the set, e.g. 0.9999 is in the set but is a distance at least 0.00001 from points outside the set.
The way these assumptions are used in section 13.3is to have a region where a function or vector field is reasonably nice and to have contours strictly inside that region--because once you let a contour get on a boundary things infinitely close might mess things up.
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