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# Manual Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus

Show that if C he. Give aD example of a. DOt exist. Urn 0 lind fA xc existe,!

Usr, Problem Let U be opcn set of Problem 3 Show that lin increa. Of sufficient importance to deserve a special designation, this theorem is usually referred to 8S Fubini's theorem, although it is more or less 8 special case of a theorem proved hy Fubini long after Theorem W8.

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J,et to,. Thus " IdlL. This will indeed turn out to be true when f is continuous, but in the general case difficulties arise. Suppose, for example, that the sct of discontinuities of f is IXol X rc,d] for some E [a, b].

The statement of Fubini's theorem therefore looks s little strange, and wi!! We will need one bit of terminology. They are called the lower and upper integrals of f on A, and denoted LI I ,. R be integrable. Now, if x E SA, then clearly ms.. We thus obtain where the proof of the last inequality is entirely analogous to t he proof of the first.

Advanced Calculus: Lecture 2 part 1: limit laws in normed linear spaces

Since J is integrable, sup! Henee 8up[L. In other words,. The assertion for '11 follows similarly from the inequalities Remarks. A similar proof shows t hat. These integral! As several problems show, the possibility of interchanging the orders of iterated integrals has many consequences. This certainly oceul"!!

The worst irrcgulB. In this case z Since f,t. There- fore h ill not integrable if h x - flJ x ,y dy is lIet equal to 0 when the integral does not 6ltill t, 5. J,bll X. Suppose, for eltarn- pie, that C ,11 X l- l,l[ - I x,. Show that A' of meMlll'fl O. I j: [a,b] X I,b] R ill continuoUB,! Uee Fubini'! Uee Fubin;', t heorem to derive an npreI! Let C be tbe eet in Problem Show that ];0. What u. DeSne F ,, Hint: F v J SS.

If f: [a,b] x e,d] Snd 0'. R- be a linear traneformation of one of the fol- lowing types. Hi,,': If deta 0, then, is the eompoeition of I ineRr trB.

## Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus

Alt A IllId 8 be Joroan. Let A J"1l "nd have the 8IIme area. Show that A lind 8 hflve the sarna volume.

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A col! Cau 1. A i8 compact Then a finite number U 1 , e'JcovcrA. It clearly suffices to construct a parti tion of unity subordinate to t he cover! We will fi rst find compact sets D, C V. The sete D, are oon- stmcted inductively as follows. I covers A. V intDtV V V. Then C U. Havi ng construct ed the seta D.. On U we can define. Case 2. For each x E A the sum ". The collection of all 'P' is the desired parti ti on of unity. CaJJe S.

Case 4. A i8 arbitrary. Let B be the union of all U in fl. By case 3 there is a par- t ition of unity for 8 ; this is also a. I An important consequence of condi tion 2 of t he theorem should be not ed. Let C C A be compact. Since C is compact, finitely many auch V z cover C. One important appli cat ion of partitions of unity will illus- trate thei r main role-piecing together results obtained locally. R is bounded in some open set around each. We definefto be integrable in the extended sense if x. This implies convergence of X. J A"" f, which we define to be f Af.

J A'". J ordan-measurabk and f i, bounded, then this defini tion of f Af agree8 with the old one. If I, and henoo of k. Finall y, this result applied to Ifl proves convergence of Xfe Of j 1- :it,. R ill a non-negative continuous function.