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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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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Geometry

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