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 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:$$\omega_{a,b,c} = \frac{(x-a)dy\wedge dz+(y-b)dz\wedge dx+(z-c)dx\wedge dy}{[(x-a)^2+(y-b)^2+(z-c)^2]^{3/2}}$$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 co-tangent 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 2-dimensional 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 Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post mathandeconstudent Geometry 0 January 28th, 2017 06:37 PM Soul Elementary Math 3 March 24th, 2016 02:41 PM BenFRayfield Applied Math 4 December 21st, 2013 10:58 PM azkuene Algebra 3 February 15th, 2012 08:13 PM dave111 Calculus 0 November 13th, 2008 03:31 PM

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