# Die Ausdehnungslehre, oder die Wissenschaft von den by Grassmann R.

By Grassmann R.

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5. Consider a differentiable map f: Rn ! R. Show that J1 f D jrfj. 6. Compute H 2 of the unit two-sphere S2 0, 1 by considering the map f: R2 ! R3 f: ϕ, Â ! sin ϕ cos Â, sin ϕ sin Â, cos ϕ . 7. Verify the coarea formula for f: R3 ! R given by f x, y, z D x 2 C y 2 C z2 , A D B3 0, R . 8. Let E be an m-dimensional rectifiable Borel subset of the unit sphere in Rn with H m E D a0 . Let C D ftx: x 2 E, 0 Ä t Ä 1g. (a) Rigorously compute H mC1 C . (b) Compute mC1 C, 0 . (c) What is TanmC1 C, 0 ? 9.

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 ! R1 such that L 1 fx: f x 6D g x g < ε, but (b) f is not Lipschitz.

S Finally, we will require that S have compact support. Such currents are called rectifiable currents. Definitions for currents are by duality with forms. The boundary of an mdimensional current T 2 Dm is the m 1 -dimensional current ∂T 2 Dm 1 defined by ∂T ϕ D T d ϕ . By Stokes’s Theorem, this agrees with the usual definition of boundary if T is (integration over) a smooth oriented manifold with boundary. 6). , ∂ ° ∂ D 0, as follows from the easy fact that d ° d. The boundary of a rectifiable current S is generally not a rectifiable current.