
Topology Topology Math Forum 
 LinkBack  Thread Tools  Display Modes 
January 28th, 2017, 06:59 PM  #1 
Newbie Joined: Jan 2017 From: Toronto Posts: 2 Thanks: 0  Calculating the size of the radial shadow that a manifold casts
I have a question about this problem and I want to understand it, but I'm not sure if my logic is solid. The question is: >Let M be compact, connected, oriented surface with boundary in $\mathbb{R}^3$, and $(a,b,c)$ not in M. Define:$$\Omega_{a,b,c} = \int_{M}\omega_{a,b,c}$$ where:$$\omega_{a,b,c} = \frac{(xa)dy\wedge dz+(yb)dz\wedge dx+(zc)dx\wedge dy}{[(xa)^2+(yb)^2+(zc)^2]^{3/2}}$$Show there is a path $j(t)$ crossing $M$ transversally, such that if $\mu$ is an orientation form on $M$ and $j'(t)\wedge \mu$ is orientation form on $\mathbb{R}^3$ (for $j'(t)$, using the metric to identify tangent and cotangent bundles), then: $$\lim_{t_{1}\rightarrow0^{}, t_2\rightarrow0^{+}}\Omega(j(t_2)j(t_1))=4\pi$$ Now reading a solution manual to Spivak's Calculus on Manifolds, I found that if $M$ is a boundary surface in $\mathbb{R}^3$ of a manifold $N$, then the integral on $M$ of $\omega_{a,b,c}$ is also $4\pi$. Now, the way I think I should approach this is by letting $N=M\cup M'$, where $M'$ is a 2dimensional Manifold. Then, let $t_2 = (a,b,c)$ and $t_1 = (a',b',c')$. Then $t_2\notin N$, $t_1 \in N\partial N$. Then: $$\Omega(j(t_2)) +\int_{M'}\omega_{a,b,c}=0,\Omega(j(t_1))+\int_{M' }\omega_{a,b,c}=4\pi$$ Subtracting these two and taking the limit as $t_2\rightarrow0^{+}, t_1\rightarrow0^{}$ gives the desired limit. Is this a fine answer for this question? Or am I missing something that could be included in this solution? Thanks for your time. 

Tags 
calculating, casts, manifold, radial, shadow, size 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Finding the Radial Shadow that a manifold casts on a sphere centered at $(a,b,c)$  mathandeconstudent  Geometry  0  January 28th, 2017 06:37 PM 
axiomatic descriptions of even size or odd size  Soul  Elementary Math  3  March 24th, 2016 02:41 PM 
Set size is relative  Same topology of negative size sets  BenFRayfield  Applied Math  4  December 21st, 2013 10:58 PM 
Inverse radial distortion  azkuene  Algebra  3  February 15th, 2012 08:13 PM 
Curl of a Radial field  dave111  Calculus  0  November 13th, 2008 03:31 PM 