By Kristopher Tapp (auth.)
This is a textbook on differential geometry well-suited to a number of classes in this subject. For readers looking an uncomplicated textual content, the must haves are minimum and comprise lots of examples and intermediate steps inside of proofs, whereas supplying a call for participation to extra excursive purposes and complicated subject matters. For readers certain for graduate institution in math or physics, it is a transparent, concise, rigorous improvement of the subject together with the deep international theorems. For the advantage of all readers, the writer employs a variety of recommendations to render the tough summary rules herein extra comprehensible and engaging.
Over three hundred colour illustrations convey the math to existence, immediately clarifying options in ways in which grayscale couldn't. Green-boxed definitions and purple-boxed theorems support to visually arrange the mathematical content material. colour is even used in the textual content to spotlight logical relationships.
Applications abound! The research of conformal and equiareal capabilities is grounded in its program to cartography. Evolutes, involutes and cycloids are brought via Christiaan Huygens' attention-grabbing tale: in trying to resolve the recognized longitude challenge with a mathematically-improved pendulum clock, he invented arithmetic that will later be utilized to optics and gears. Clairaut’s Theorem is gifted as a conservation legislations for angular momentum. Green’s Theorem makes attainable a drafting instrument known as a planimeter. Foucault’s Pendulum is helping one visualize a parallel vector box alongside a range of the earth. Even larger, a south-pointing chariot is helping one visualize a parallel vector box alongside any curve in any surface.
In fact, the main profound program of differential geometry is to trendy physics, that is past the scope of this e-book. The GPS in any motor vehicle wouldn’t paintings with out common relativity, formalized during the language of differential geometry. all through this publication, purposes, metaphors and visualizations are instruments that encourage and make clear the rigorous mathematical content material, yet by no means substitute it.
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Extra resources for Differential Geometry of Curves and Surfaces
0), . , en = (0, 0, . , 0, 1)}. Proof. (1) ⇐⇒ (2) If LA is a rigid motion, then for all p ∈ Rn , |LA (p)| = |LA (p) − 0| = |LA (p) − LA (0)| = |p − 0| = |p|, so A is orthogonal. Conversely, if A is orthogonal, then for all p, q ∈ Rn , |LA (p) − LA (q)| = |LA (p − q)| = |p − q|, so LA is a rigid motion. (2) ⇐⇒ (3) If (3) is true, then for all p ∈ Rn , we have |LA (p)| = LA (p), LA (p) 1 2 = p, p 1 2 = |p|, so (2) is true. In other words, if LA preserves inner products, then it preserves norms.
Notice that g is a rigid motion with g(0) = 0 (so g preserves norms and inner products, as above) and g(ei ) = ei for all i = 1, . , n. Let p ∈ Rn . Write p = ai ei and g(p) = bi ei . Then, bi = g(p), ei = g(p), g(ei ) = p, ei = ai , which proves g(p) = p, so g is the identity function. What about rigid motions that do not ﬁx the origin? For any q ∈ Rn , the translation map Tq : Rn → Rn , deﬁned as Tq (p) = p + q, is clearly a rigid motion that sends the origin to q. We’ll prove next that there are no rigid motions other than compositions of translations with linear rigid motions.
Proof. Deﬁne q = f (0). Notice that (Tq )−1 ◦ f is a rigid motion that ﬁxes the origin. 57, (Tq )−1 ◦ f = LA for some A ∈ O(n). Thus, f = Tq ◦ LA . The uniqueness claim is left to the reader. 59. If A ∈ O(n), then det(A) = 1 or det(A) = −1. Proof. Since AT · A = I, 2 1 = det(I) = det(AT · A) = det(AT ) · det(A) = (det(A)) . 60. 58) is called proper if det(A) = 1, and improper if det(A) = −1. 61. If A ∈ O(2), then its two columns form an orthonormal basis of R2 . Its ﬁrst column is an arbitrary unit-length vector of R2 , which can be written as x = (cos θ, sin θ) for some θ ∈ [0, 2π).