By Frank Morgan

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Within the Spring of 1966, I gave a chain of lectures within the Princeton collage division of Physics, geared toward contemporary mathematical leads to mechanics, particularly the paintings of Kolmogorov, Arnold, and Moser and its software to Laplace's query of balance of the sunlight procedure. Mr. Marsden's notes of the lectures, with a few revision and growth by means of either one of us, grew to become this publication.

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**Example text**

Let W be an mdimensional rectifiable subset of Rn , Z a -dimensional rectifiable subset of R , m ½ ½ 1, and f a Lipschitz function from W to Z. Then ap J fd H m D W Hm f 1 fzg d H z. Z More generally, for any H m W integrable function g on W, g Ð ap J fd H m D W g dH m z f 1 fzg d H z. Lipschitz Functions and Rectifiable Sets 31 Note: If f has an extension f to Rn , ap J f Ä J f (where both are defined). 14. 23]. Let W be an m-dimensional rectifiable Borel subset of Rn and let Z be a -dimensional rectifiable Borel subset of R .

For the rectifiable currents T 2 R1 R2 and for C1 functions f, g, and h, compute formulas for T f d x C g dy and ∂T h : (a) T D H 1 f x, 0 : 0 Ä x Ä 1g ^ e1 . p (b) T D H 1 f x, x : 0 Ä x Ä 1g ^ 3 2 e1 C e2 . 10. Prove that Im is M dense in Rm and F dense in Fm . 11. Prove that fT 2 Rm : spt T ² B 0, R g is M complete and that fT 2 Fm : spt T ² B 0, R g is F complete. 12. Prove that ∂ carries Nm into Nm 1 and Fm into Fm 1 . 13. 3(1): Nm D fT 2 Rm : M ∂T < 1g, Rm D fT 2 Fm : M T < 1g. 14. Prove that, in analogy with the definitions of Im and Fm , Nm D fT 2 Rm : ∂T 2 Rm 1 g, Fm D fT C ∂S: T 2 Rm , S 2 RmC1 g.

1. A purely unrectifiable one-dimensional set E. E is invisible from almost all directions. slope 12 ) gives an exceptional case: the projection is a solid interval. If A is any rectifiable set, then H 1 A \ E D 0. 1. Give an example of a Lipschitz function f: [0, 1] ! R such that f is not differentiable at any rational point. 2. 3 to deduce that a Lipschitz function is approximately differentiable almost everywhere. 3. Give an example of a continuous function f: R ! R such that (a) given ε > 0 there is a C1 function g: R1 !