This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ...
View Full
Document
 Spring '10
 Wong
 Geometry

Click to edit the document details