偏微分方程 第1卷

i differentiation and integration on manifolds
  §1 the weierstraβ approximation theorem
  §2 parameter-invariant integrals and differential forms
  §3 the exterior derivative of differential forms
  §4 the stokes integral theorem for manifolds
  §5 the integral theorems of gaub and stokes
  §6 curvilinear integrals
  §7 the lemma of poineare
  §8 co-derivatives and the laplace-beltrami operator
  §9 some historical notices to chapter i
ii foundations of functional analysis
  §1 daniell's integral with examples
  §2 extension of daniell's integral to lebesgue's integral
  §3 measurable sets
  §4 measurable functions
  §5 riemann's and lebesgue's integral on rectangles
  §6 banach and hilbert spaces
  §7 the lebesgue spaces lp(x)
  §8 bounded linear funetionals on lp(x) and weak convergence
  §9 some historical notices to chapter ii
iii brouwer's degree of mapping with geometric applications
  §1 the winding number
  §2 the degree of mapping in rn
  §3 geometric existence theorems
  §4 the index of a mapping
  §5 the product theorem
  §6 theorems of jordan-brouwer
iv generalized analytic functions
  §1 the cauchy-riemann differential equation
  §2 holomorphic functions in cn
  §3 geometric behavior of holomorphic functions in c
  §4 isolated singularities and the general residue theorem
  §5 the inhomogeneous cauchy-riemann differential equation
  §6 pseudoholomorphic functions
  §7 conformal mappings
  §8 boundary behavior of conformal mappings
  §9 some historical notices to chapter iv
v potential theory and spherical harmonics
  §1 poisson's differential equation in rn
  §2 poisson's integral formula with applications
  §3 dirichlet's problem for the laplace equation in rn
  §4 theory of spherical harmonics: fourier series
  §5 theory of spherical harmonics in n variables
vi linear partial differential equations in rn
  §1 the maximum principle for elliptic differential equations
  §2 quasilinear elliptic differential equations
  §3 the heat equation
  §4 characteristic surfaces
  §5 the wave equation in rn for n = 1, 3, 2
  §6 the wave equation in rn for n _2
  §7 the inhomogeneous wave equation and an initial-boundary-value problem
  §8 classification, transformation and reduction of partialdifferential equations
  §9 some historical notices to the chapters v and vi
references
index