By Frank Morgan

Geometric degree thought is the mathematical framework for the examine of crystal development, clusters of cleaning soap bubbles, and comparable buildings concerning minimization of power. Morgan emphasizes geometry over proofs and technicalities, and contains a bibliography and plentiful illustrations and examples. This **Second Edition** encompasses a new bankruptcy on cleaning soap bubbles in addition to up-to-date sections addressing quantity constraints, surfaces in manifolds, unfastened limitations, and Besicovitch consistent effects. The textual content will introduce beginners to the sector and attract mathematicians operating within the box

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**Example text**

Then th T(fjeJ) = ( r L / ) ( e y ) « (TLfj){p*ej) = (p (TL/ ))(e*). y y# y Since p (Tlfj) G F^(R ) is of the form & Α ξ by part II, and its support has measure 0, it must be 0. Therefore r(/ e*) = 0, as desired. 8. 23] . Given a real flat chain Τ G F and ε > 0, there is a real polyhedral approximation Ρ G P satisfying ¥(T — Ρ) < ε and Μ(Ρ)<Μ(Γ)+ε. 6), we may assume Τ G N . By smoothing (cf. 7, part I), we may assume Τ is of the form Τ = ^f Α ξ(χ), where £(x) is a smooth m-vectorfield of compact support with f\£(x)\dJ*?

M n l n Proof CASE J f=l, n 1. fis orthogonal projection. If / is orthogonal projection, then and the coarea formula is reduced to Fubini's Theorem. GENERAL CASE. We treat just the main case J f Φ 0. By subdividing A as in the proof of the area formula, we may assume that / is linear. Then f = L P, where Ρ denotes projection onto the n-dimensional orthogonal complement V of the kernel of / and where L is a a nonsingular linear map from V to R . Now n Q n ( J fd& J A n m = \detL\

2. 2. Although a rectifiable set Ε must have finite area, its boundary can wiggle enough to have infinite length. Thus a rectifiable current need not be an integral current. Here the length of each successive smaller square rapidly approaches one-third the length of the larger square. 3. This infinite collection of discs gives another example of a rectifiable current which is not an integral current. There is finite total area, but infinite total boundary length. Alternatively, let Ε be a countable union of discs of radius 1/k: E= (J {{x,y,z):x + y