# Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus

## Michael Spivak

Language: English

Pages: 160

ISBN: 0805390219

Format: PDF / Kindle (mobi) / ePub

The A to Z of Mathematics: A Basic Guide

Introduction to the Geometry of Complex Numbers (Dover Books on Mathematics)

Solutions Manual to Accompany Ordinary Differential Equations

Handbook of Combinatorial Designs (2nd Edition) (Discrete Mathematics and Its Applications)

Problem-Solving Strategies (Problem Books in Mathematics)

An Introduction to Difference Equations (3rd Edition) (Undergraduate Texts in Mathematics)

such that the matrix (Djfi(a)) 1 ≤ i ≤ p, j = j1, . . ,jp has non-zero determinant. If g : Rn → Rn permutes the xj so that g(x1, . . . ,xn) = ), then f ◦ g is a function of the type already considered, so ((f ◦ g) ◦ k)(x1, . . . ,xn) = (xn—p+1, . . . ,xn) for some k. Let h = g ◦ k. Problems. 2-40. Use the implicit function theorem to re-do Problem 2-15(c). 2-41. Let f: R × R → R be differentiable. For each x ∈ R define gx: R → R by gx(y) = f(x,y). Suppose that for each x there is a

P(x,y) = (r(x,y), θ(x,y)), where (Here arctan denotes the inverse of the function tan: ( – π/2,π/2) → R.) Find Pʹ(x,y). The function P is called the polar coordinate system on A. (c) Let C ⊂A be the region between the circles of radii r1 and r2 and the half-lines through 0 which make angles of θ1 and θ2 with the x-axis. If h: C → R is integrable and h(x,y) = g(r(x,y), θ(x,y)), show that If Br = {(x,y): x2 + y2 ≤ r2}, show that (d) If Cr = [ – r,r] × [ – r,r], show that and (e) Prove that

M1,M2 are compact, prove that where ω is an (n—1)-form on M1, and ∂M1 and ∂M2 have the orientations induced by the usual orientations of M1 and M2. Hint: Find a manifold-with-boundary M such that ∂M = ∂M1 ∪ ∂M2 and such that the induced orientation on ∂M agrees with that for ∂M1 on ∂M1 and is the negative of that for ∂M2 on ∂M2. THE VOLUME ELEMENT Let M be a k-dimensional manifold (or manifold-with-boundary) in Rn, with an orientation �. If x ∈ M, then �x and the inner product Tx we defined

volume of Sn-1 = {x ∈ Rn: |x| = 1} in terms of the n-dimensional volume of Bn = {x ∈ Rn: |x| ≤ 1}. (This volume is πn/2(n/2)! if n is even and 2(n+1)/2π(n – 1)/2/1. 3 · 5 · . . . . n if n is odd.) 5-36. Define F on R3 by F(x) = (0,0,cx3)x, and let M be a compact three-dimensional manifold-with-boundary with M ⊂ {x: x3 ≤ 0}.The vector field F may be thought of as the downward pressure of a fluid of density c in {x: x3 ≤ 0). Since a fluid exerts equal pressures in all directions, we define the

1-27. Prove that {x ∈ R n: ∣x – a∣ < r} is open by considering the function f: Rn → R with f(x) = ∣x – a∣. 1-28. If A ⊂ R n is not closed, show that there is a continuous function f: A → R which is unbounded. Hint: If x ∈ R n – A but x ∉ interior ( R n – A), let f(y) = 1/∣y – x∣. 1-29. If A is compact, prove that every continuous function f: A → R takes on a maximum and a minimum value. 1-30. Let f: [a,b] → R be an increasing function. If x1, . . . ,xn ∈ [a,b] are distinct, show that