By Kristopher Tapp (auth.)

This is a textbook on differential geometry well-suited to numerous classes in this subject. For readers looking an hassle-free textual content, the necessities are minimum and contain lots of examples and intermediate steps inside proofs, whereas supplying a call for participation to extra excursive functions and complicated themes. For readers certain for graduate tuition in math or physics, this can be a transparent, concise, rigorous improvement of the subject together with the deep international theorems. For the good thing about all readers, the writer employs a variety of innovations to render the tough summary rules herein extra comprehensible and engaging.

Over three hundred colour illustrations carry the math to existence, immediately clarifying techniques in ways in which grayscale couldn't. Green-boxed definitions and purple-boxed theorems aid to visually set up the mathematical content material. colour is even used in the textual content to spotlight logical relationships.

Applications abound! The examine of conformal and equiareal features is grounded in its program to cartography. Evolutes, involutes and cycloids are brought via Christiaan Huygens' interesting tale: in trying to clear up the recognized longitude challenge with a mathematically-improved pendulum clock, he invented arithmetic that may 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 software referred to as a planimeter. Foucault’s Pendulum is helping one visualize a parallel vector box alongside a range of the earth. Even higher, 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 ebook. The GPS in any vehicle wouldn’t paintings with out normal relativity, formalized during the language of differential geometry. all through this e-book, functions, metaphors and visualizations are instruments that inspire and make clear the rigorous mathematical content material, yet by no means change it.

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**Sample text**

Deﬁne v(t) = γ (t) and a(t) = γ (t) as before. It does not quite work to deﬁne κ(t) to equal |a⊥ (t)|, because this deﬁnition would depend on the parametrization. We encountered this conceptually at the end of Sect. 3, when we mentioned that |a⊥ (t)| depends not only on how sharply the path bends, but also on the object’s speed, so it would be increased by a faster ˜ = γ ◦ φ be parametrization. We can now quantify this dependence. Let γ a reparametrization of γ, whose velocity and acceleration functions will be ˜ (t) and a ˜ (t) respectively.

If γ : I → R2 is a regular plane curve (not necessarily parametrized by arc length), then for all t ∈ I, a(t), R90 κs (t) = v(t) |v(t)| |v(t)|2 = a(t), R90 (v(t)) . |v(t)|3 This deﬁnition of κs (t) agrees with our previous deﬁnition for the special case in which γ is of unit speed. 50). It follows that |κs (t)| = κ(t) even for non-unit-speed curves. This could also be veriﬁed by ⊥ | comparing the deﬁnition κ(t) = |a |v|2 to the above formula for κs (t), noticing that a(t), R90 v(t) |v(t)| = ±|a⊥ |.

Let γ : I → R3 be a regular space curve with κ(t) = 0 for all t ∈ I. The trace of γ is constrained to a plane if and only if τ (t) = 0 for all t ∈ I. 7. SPACE CURVES 45 Proof. First suppose that the trace of γ is constrained to the plane, P , with equation ax + by + cz = d. Let w = (a, b, c) ∈ R3 , and notice that this plane can be redescribed as P = {p ∈ R3 | p, w = d}. Since γ(t), w = d is a constant function of t, its derivatives vanish: 0= d γ(t), w = v(t), w , dt 0= d2 γ(t), w = a(t), w . dt2 It follows from this that t(t) and n(t) are both orthogonal to w, so their cross w .