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Definition
a subset S in R3, for each p in S, there exists a Neighborhood V in R3 s.t.
1. X is differentiable
2. x is a homeomorphism
3. for all q in U, the differential dx_q: R2 -> R2 is 1-1 |
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| there exists x^-1: V int S -> U which is continuous |
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| if f is differentiable, (x,y,f(x,y)) is what? |
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Definition
dF_p:Rn->Rm is not a surjective (onto) mapping.
f'(x_0) = 0
the image F(p) in Rm of a crit point |
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| If f is D'able, and a in f(U) is a regular surface, then... |
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| f^-1(a) is a regular surface |
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| If x is 1-1, p in a regular surface, and holds conditions 1 + 3, then.. |
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| Change of parameters is a... |
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| there exists a differentiable map Q:S1->S2 with a differentiable inverse Q^-1: S2->S1 |
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| subset of C in R3 s.t. for all p in C, there exists a Neighborhood V of p in R3 and a d able homeomorphism alpha:I in R -> V int C s.t. d'alpha is 1-1 for all t in I. |
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| dx_q R2->R3 is 1-1 for all q in U |
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| if x is a regular parametrized surface, and q in U, then... |
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| there exists a Neighborhood V of q in R2 s.t. x(V) in R3 is a regular surface |
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| I_p on T_p(s) is called the 1st fundamental form of the regular surface s in R3 at p in S s.t. ... |
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| "I_p (w) = _p = |w|^2 >= 0" |
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Definition
| E*(u')^2 + 2 F*u'v' + G(v')^2 |
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| Arc length of a parametrized a:I->S = |
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Definition
s(t) = int(|a'(t)|dt, 0,t)
= int((I(a'(t))^(1/2))dt, 0,t)
if u, v apply...
= Int((E(u')^2 + 2*F*u'*v' + G(v')^2)^(1/2)dt,0,t) |
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Definition
=Int(Int( |x_u V x_v| du,dv))_Q
,Q = x^-1(R) |
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| (x_u V x_v) / |x_u V x_v| (q), q in x(u) |
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II_p in T_p(s) =
(Second Fundamental Form of S at p) |
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normal curvature of C in S at p,
K_n= |
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=K * cos (theta),
for cos(theta) = ,
n = normal vector to C
N = normal vector to S |
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| for all alpha(s) in S, @ p, having the same tangent line... |
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| have the same normal curvatures at this point |
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Definition
1 - maximum normal curvature
2 - minimum are principal curvatures
e_1, e_2 are their directions called principal directions @ p |
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Definition
"E = < x_u, x_u >
F = < x_u, x_v >
G = < X_v, X_v >" |
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