推 sulanpa :感謝! 12/07 12:56
※ 引述《sulanpa (...)》之銘言：
: For each of following maps f: R^2 → R^3, describe the surface S = f(R^2) and
: find a description of S as the locus of an equation F(x,y,z) = 0. Find the
: points where (p_u)f and (p_v)f are linearly dependent, and describe the
: singularities of S(if any) at these points. f(u,v) = (aucosv,businv,u) (a,b>0)
: (p 是偏微分符號)
: 謝謝
Let x=aucosv,y=businv,z=u---(*). Then (x,y,z) is in S and
(x/a)^2+(y/b)^2-z^2
=(u^2)(cosv)^2+(u^2)(sinv)^2-u^2
=u^2-u^2=0.
Conversly, we can also find a point (u,v) s.t. the above equalities (*) hold.
So define F(x,y,z)=(x/a)^2+(y/b)^2-z^2. Then every point in S must satisfy
F(x,y,z)=0.
For all (u,v), (p_u)f(u,v)=(acosv,bsinv,1) and (p_v)f(u,v)=(-ausinv,bucosv,0).
(p_u)f and (p_v)f are linearly dependent
<=> s*(p_u)f+t*(p_v)f≡0 for some nonzero vector (s,t)
<=> ascosv=atusinv, bssinv=-btucosv, s=t*0=0
<=> atusinv=-btucosv=0
<=> ausinv=-bucosv=0 [(s,t) is nonzero => t≠0]
If u=0, then the equality above holds; otherwise, sinv=cosv=0.
This is impossible. So these points are {(0,v)|v is a real number}.
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律:知道嗎?聽說我們的歌被海外的電視台所錄用耶!看來我們離武道館不遠了
唯:真的嗎?那真的是太好了，我一直夢想能在武道館彈著吉太，好高興
紬:小唯能高興真的是太好了，呵呵~
澪:拜託!那個明明是盜用不是錄用，你們怎麼還這麼高興?
律、唯、紬:啊?什麼? 輕音部
澪:絕望啦!我對盜用錄用分不清楚的輕音部社員們絕望啦! 邁向武道館之路
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