differential geometry w notes from teacher_Part_71

differential geometry w notes from teacher_Part_71 - 141...

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Unformatted text preview: 141 5.4. DEGREE OF A MAP • Let ϕ : T → S 2 be a smooth map defined by ϕ(θ) = x1 (θ1 ) − x2 (θ2 ) . ||x1 (θ1 ) − x2 (θ2 )|| • The Gauss linking number of the loops C1 and C2 is defined by Link(C1 , C2 ) = deg(ϕ) . • Problem. Let x12 = x2 − x1 . Show that 1 4π 1 = 4π 1 Link(C1 , C2 ) = C1 2π x 3 12 C2 ||x12 || 2π dθ1 dθ2 0 0 × dx12 · dx1 ˙ ˙ {[x2 (θ2 ) − x1 (θ1 )] × x2 (θ2 )} · x1 (θ1 ) 3 ||x2 (θ2 ) − x1 (θ1 )|| • Let V be an orientable surface in R3 such that ∂V = C1 . • Let N be the normal vector to V consistent with the orientation of C1 . • Let the curve C2 intersect V transversally. • The intersection number V ◦ C2 of the curve C2 and the surface V is the signed number of intersections of C2 and V , with an intersection being positive if the tangent vector to C2 at the point of the intersection has the same direction as N, that is, V ◦ C2 = ˙ sign (N, x) x , i x i ∈V where the sum goes over all intersection points xi . • Problem. Show that Link(C1 , C2 ) = V ◦ C2 . diffgeom.tex; April 12, 2006; 17:59; p. 141 142 CHAPTER 5. LIE DERIVATIVE diffgeom.tex; April 12, 2006; 17:59; p. 142 ...
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differential geometry w notes from teacher_Part_71 - 141...

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