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《偏微分方程(第1捲)》是一部兩捲集的偏微分方程教材。多變量橢圓，拋物和雙麯方程是研究的主要對象，解決瞭PDE和多變量方法之間的關係。第一捲中集中研究瞭流形上的積分和微分，泛函解析基礎，映射的Brouwer度，廣義解析函數和圓周同調這些議題，在這一捲中通過積分錶示論解決偏微分方程問題，第二捲中講述函數解析解法。書中各章的獨立性較強，有一定偏微分方程基本知識的讀者可以獨立閱讀各章。

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
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