10 CONTENTS

29 The Area Formula 100929.1 Estimates for Hausdorff Measure . . . . . . . . . . . . . . . . . . . . . .100929.2 Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . .101129.3 A Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101229.4 Estimates and a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . .101429.5 The Area Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101829.6 Mappings that are not One to One . . . . . . . . . . . . . . . . . . . . . .102129.7 The Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .102329.8 The Reynolds Transport Formula . . . . . . . . . . . . . . . . . . . . . .103129.9 The Coarea Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103429.10 Change of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104529.11 Integration and the Degree . . . . . . . . . . . . . . . . . . . . . . . . . .104729.12 The Case Of W 1,p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1050

30 Integration Of Differential Forms 106130.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106130.2 The Binet Cauchy Formula . . . . . . . . . . . . . . . . . . . . . . . . .106330.3 Integration Of Differential Forms On Manifolds . . . . . . . . . . . . . .1064

30.3.1 The Derivative Of A Differential Form . . . . . . . . . . . . . .106830.4 Stoke’s Theorem And The Orientation Of ∂Ω . . . . . . . . . . . . . . .106930.5 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1074

30.5.1 An Oriented Manifold . . . . . . . . . . . . . . . . . . . . . . .107430.5.2 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .1076

30.6 The Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .1076

31 Differentiation, Radon Measures 108131.1 Fundamental Theorem Of Calculus . . . . . . . . . . . . . . . . . . . . .108131.2 Slicing Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .108431.3 Differentiation of Radon Measures . . . . . . . . . . . . . . . . . . . . .1090

31.3.1 Radon Nikodym Theorem for Radon Measures . . . . . . . . . .1093

32 Fourier Transforms 109732.1 An Algebra Of Special Functions . . . . . . . . . . . . . . . . . . . . . .109732.2 Fourier Transforms Of Functions In G . . . . . . . . . . . . . . . . . . .109832.3 Fourier Transforms Of Just About Anything . . . . . . . . . . . . . . . .1100

32.3.1 Fourier Transforms Of G ∗ . . . . . . . . . . . . . . . . . . . . .110032.3.2 Fourier Transforms Of Functions In L1 (Rn) . . . . . . . . . . .110432.3.3 Fourier Transforms Of Functions In L2 (Rn) . . . . . . . . . . .110732.3.4 The Schwartz Class . . . . . . . . . . . . . . . . . . . . . . . .111132.3.5 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . .1113

32.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1115

33 Fourier Analysis In Rn 111933.1 The Marcinkiewicz Interpolation Theorem . . . . . . . . . . . . . . . . .111933.2 The Calderon Zygmund Decomposition . . . . . . . . . . . . . . . . . .112233.3 Mihlin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1123