多元微積分

內(nèi)容概要

　　《多元微積分(第3版)》是全面，知識(shí)體系新穎的多變量微積分教程。旨在解決廣大多變量微積分學(xué)者遇到的新老問(wèn)題，包內(nèi)包括：（第一部分）基礎(chǔ)資料：向量；向量微分；多變量函數(shù)；鏈?zhǔn)椒▌t和梯度；（第二部分）最大值，最小值和泰勒公式：最大值和最小值；高階導(dǎo)數(shù)；（第三部分）曲線積分和雙積分：勢(shì)函數(shù)；曲線積分；雙積分；格林定理；（第四部分）三重積分和曲面積分：三重積分；曲面積分；（第五部分）映射、反映射和變量變換公式：矩陣；線性映射；行列式；多變量函數(shù)的應(yīng)用；變量變換公式；附錄：傅里葉級(jí)數(shù)。
　　讀者對(duì)象：數(shù)學(xué)專業(yè)的本科生、研究生和相關(guān)專業(yè)的數(shù)學(xué)工作者。

書(shū)籍目錄

part one basic material
　chapter i vectors
　　1. definition of points in space
　　2. located vectors
　　3. scalar product
　　4. the norm of a vector
　　5. parametric lines
　　6. planes
　　7. the cross product
　chapter ii differentiation of vectors
　　1. derivative
　　2. length of curves
　chapter iii functions of several variables
　　1. graphs and level curves
　　2. partial derivatives
　　3. differentiability and gradient
　　4. repeated partial derivatives
　chapter iv the chain rule and the gradient
　　1. the chain rule
　　2. tangent plane
　　3. directional derivative
　　4. functions depending only on the distance from the origin
　　5. the law of conservation of energy
　　6. further technique in partial differentiation
part two maxima, minima, and taylor's formula
　chapter v maximum and minimum
　　1. critical points
　　2. boundary points
　　3. lagrange multipliers
　chapter vi higher derivatives
　　1. the first two terms in tayior's formula
　　2. the quadratic term at critical points
　　3. algebraic study of a quadratic form
　　4. partial differential operators
　　5. the general expression for tayior's formula
　　appendix. taylor's formula in one variable
　　note. chapter ix on double integrals is self contained, and could
be covered here.
part three curve integrals and double integrals
　chapter vii potential functions
　　1. existence and uniqueness of potential functions
　　2. local existence of potential functions
　　3. an important special vector field
　　4. differentiating under the integral
　　5. proof of the local existence theorem
　chapter viii curve integrals
　　1. definition and evaluation of curve integrals
　　2. the reverse path
　　3. curve integrals when the vector field has a potential
function
　　4. dependence of the integral on the path
　chapter ix double integrals
　　1. double integrals
　　2. repeated integrals
　　3. polar coordinates
　chapter x green's theorem
　　1. the standard version
　　2. the divergence and the rotation of a vector field
part four triple and surface integrals
　chapter xi triple integrals
　　1. triple integrals
　　2. cylindrical and spherical coordinates
　　3. center of mass
　chapter xii surface integrala
　　1. parametrization, tangent plane, and normal vector
　　2. surface area
　　3. surface integrals
　　4. cuff and divergence of a vector field
　　5. divergence theorem in 3-space
　　6. stokes' theorem
part five mappings, inverse mappings, and change of variables
formula
　chapter xiii matrices
　　1. matrices
　　2. multiplication of matrices
　chapter xiv linear mappings
　　1. mappings
　　2. linear mappings
　　3. geometric applications
　　4. composition and inverse of mappings
　chapter xv determinants
　　1. determinants of order 2
　　2. determinants of order 3
　　3. additional properties of determinants
　　4. independence of vectors
　　5. determinant of a product
　　6. inverse of a matrix
　chapter xvi applications to functions of several variables
　　1. the jacobian matrix
　　2. differentiabiiity
　　3. thc chain rule
　　4. invcrse mappings
　　5. implicit functions
　　6. the hessian
　chapter xvii the change of variables formula
　　1. determinants as area and volume
　　2. dilations
　　3. change of variables formula in two dimensions
　　4. application of green's formula to the change of variables
formula
　　5. change of variables formula in three dimensions
　　6. vector fields on the sphere
appendix fourier series
　1. general scalar products
　2. computation of fourier series
answers
index

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