cobble
发表于 2025-3-21 16:52:11
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MIRE
发表于 2025-3-21 20:43:27
The Gauss Map,on .:. → ℝ.. associated with the vector field . by .(.) = (., .(.)), . ∈ ., actually maps . into the unit .-sphere S. ⊂ ℝ.. since ∥.(.)∥ = 1 for all . ∈ .. Thus, associated to each oriented .-surface . is a smooth map .: . → S.. called the .. . may be thought of as the map which assigns to each poin
ENNUI
发表于 2025-3-22 02:57:24
Geodesics, proccss of differentiation of vector fields and functions defined along parametrized curves. In order to allow the possibility that such vector fields and functions may take on different values at a point where a parametrized curve crosses itself, it is convenient to regard these fields and functio
Coronation
发表于 2025-3-22 06:30:25
Parallel Transport,however, generally not tangent to .. We can, nevertheless, obtain a vector field tangent to . by projecting Ẋ(.) orthogonally onto .. for each . ∈ . (see Figure 8.1). This process of differentiating and then projecting onto the tangent space to . defines an operation with the same properties as diff
Gum-Disease
发表于 2025-3-22 10:31:07
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金桌活画面
发表于 2025-3-22 13:27:56
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金桌活画面
发表于 2025-3-22 20:06:03
Convex Surfaces,ee Figure 13.1). An oriented .-surface . is . at . ∈ . if there exists an open set . ⊂ ℝ.. containing . such that . ∩ . is contained either in . or in .. Thus a convex .-surface is necessarily convex at each of its points, but an .-surface convex at each point need not be a convex .-surface (see Fig
languid
发表于 2025-3-22 23:24:59
Parametrized Surfaces,ure . and (ii) define various integrals over .. We shall now carry out a similar program for .-surfaces (. > 1). It will turn out that oriented .-surfaces (even connected ones) in general admit only local parametrizations, but that will be adequate for our needs.
湿润
发表于 2025-3-23 02:04:56
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阴郁
发表于 2025-3-23 06:50:14
The Exponential Map, We begin by using a technique of the calculus of variations analogous to the one we used in Chapter 18 to study minimal surfaces. Now, however, we shall vary parametrized curves rather than parametrized surfaces