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Partial Fraction Decomposition

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If and

then can be written in the form

.


The Laplace Expansion for Determinants

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(moved to Laplace expansion article)

Calculus on Manifolds

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2-4 Suppose that f is differentiable at (0,0) and let λ = Df(0,0). Then

, or
, so that in particular
so that λ(1,0) = 0, and similarly
so that λ(0,1) = 0, hence λ = 0 and
or
, hence
; but

for all t ≠ 0, so

for all x ≠ 0.


Rectangles!

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Definition. An open rectangle in Rn is a set of the form , given real numbers . Notice that the empty set is an open rectangle. A closed rectangle is defined similarly by replacing < with ≤. The term rectangle means either an open or a closed rectangle.

Definition. The elementary volume of a rectangle R defined as above is . It follows that .

Definition. A partition of a rectangle in Rn is a finite collection of pairwise-disjoint open subrectangles of such that the union of their closures is the closure of .


Lemma. Suppose is a finite collection of rectangles and is a finite collection of hyperplanes parallel to an axis. Then determines a unique partition of each rectangle in .

Proof: It suffices to prove the lemma in the case of one rectangle and one hyperplane, and in this case the statement is obvious.

Lemma. Suppose A and B are rectangles and B is a subset of A. Then there exists a partition of A that contains B.

Proof: Let H be the collection of the 2n hyperplanes coinciding with the boundary planes of B. By the first lemma H determines a unique partition of A, and by construction B is an element of the partition.

Lemma. Suppose is a partition of a rectangle R and the open rectangle A is a subset of R. Then the collection is a partition of A.

Proof: We must show that . First we show that , which will follow from since then . Suppose then that x is in A. Then x is in R and hence is in for some p in . If x is in p then it's also in , hence in D. If x is on the boundary of p and N is any neighbourhood of x, then is a neighbourhood of x and so contains a point of p. Hence every neighbourhood of x contains a point of , i.e. x is in and so in D.

On the other hand, if y is in D, y is in for some p so it can't be exterior to A. In other words, . Thus, .


Definition. A partition is a refinement of a partition if every member of is contained in a member of .

Definition. The common refinement of the partitions and of a rectangle is the collection .

Proposition. The common refinement of the partitions and of a rectangle R is a partition of R.

Proof: Clearly the members of the common refinement are pairwise-disjoint open subrectangles of R. If x is in the closure of R then x is in for some p in . By the previous lemma, induces a partition on p, i.e. there is some p' in such that contains x.


Theorem. Suppose are pairwise-disjoint open rectangles, and are open rectangles such that . Then .

Proof: Let be a rectangle containing the . Then by the second lemma there exists, for each j, a partition of containing . Let be the common refinement of these partitions. Since refines , for each p in there exists an element of containing p; hence p is either a subset of or disjoint from it. Thus , the characteristic function of , is a well-defined step function on . Moreover, by the third lemma the elements of contained in form a partition of , and

.

Now if is the characteristic function of then because the are pairwise-disjoint. Hence . And since , it is also true that ; that is,

.

Second proof: Let and be the characteristic functions of and . Then and , and since , it is also true that . Thus

.