Download A First Course in Geometric Topology and Differential by Ethan D. Bloch PDF

By Ethan D. Bloch

The distinctiveness of this article in combining geometric topology and differential geometry lies in its unifying thread: the inspiration of a floor. With quite a few illustrations, workouts and examples, the scholar involves comprehend the connection among smooth axiomatic process and geometric instinct. The textual content is saved at a concrete point, 'motivational' in nature, averting abstractions. a couple of intuitively beautiful definitions and theorems pertaining to surfaces within the topological, polyhedral, and delicate circumstances are provided from the geometric view, and aspect set topology is specific to subsets of Euclidean areas. The remedy of differential geometry is classical, facing surfaces in R3 . the cloth here's available to math majors on the junior/senior point.

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Extra info for A First Course in Geometric Topology and Differential Geometry

Example text

We get = f dx, if/ = g dy. Using the formula for d computed above, t As usual, multiplication takes precedence over addition or subtraction, so this expression should be read as (d A yf) — ( A d\p). [Chap. I CALCULUS ON EUCLIDEAN SPACE 30 d A ^ = (j- dy dx + ^dz dxj * g dy = 0 + — g dz dx dy = g — dx dy dz. dy dz Similarly, <> / A dì// = f dx A ( — dx dy + -^ dz dy) \dx dz / = Jf — dx dz dy = —f J J — dx dy u dz. dz dz Thus d A xp - A dxf/ = (g ■£ + -£fj dx dy dz. But A \p = fg dx dy, so we get d( A xp) = d(fg) dx dy = ^^- dz dx dy (%>+'%)dx dy dz.

Let the (a) (b) (c) (d) y be a vector field on the helix a(t) = (cos t, sin t, t). I n each of following cases, express Y in the form ^ y%Ui: Y(t) is the vector from a(t) to the origin of E 3 . Y{t) = a(t) a"(t). Y(t) has unit length and is orthogonal to both a (t) and a"(t). Y(t) is the vector from a(t) to a(t -f TT). 56 FRAME FIELDS [Chap. II 7. Let F be a vector field on a curve a. If a(h) is a reparametrization of a, show that Y(h) is a vector field on a(h), and prove the chain rule Y(h)' = ti Y'(h).

I CALCULUS ON EUCLIDEAN SPACE 30 d A ^ = (j- dy dx + ^dz dxj * g dy = 0 + — g dz dx dy = g — dx dy dz. dy dz Similarly, <> / A dì// = f dx A ( — dx dy + -^ dz dy) \dx dz / = Jf — dx dz dy = —f J J — dx dy u dz. dz dz Thus d A xp - A dxf/ = (g ■£ + -£fj dx dy dz. But A \p = fg dx dy, so we get d( A xp) = d(fg) dx dy = ^^- dz dx dy (%>+'%)dx dy dz. Hence the formula is proved in this case. Case 3. The general case. From cases 1 and 2 we know that the formula is true whenever <> / and \p are "simple," that is, of the form / du, where u is x, y, or z.

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