第八章 多元函数微分法及其应用Chapter8 Differentiation of Functions of Several Variables and Its Application8.1多元函数的基本概念（The Basic Concepts of Functions of Several Variables）定义1 设D是的一个非空子集，称映射为定义在D上的二元函数，通常记为或。其中点集D称为该函数的定义域，、称为自变量，称为因变量。Definition 1 Let D be a nonempty subset of ,we call the mapping the function of two variables defined on ,usually denoted by ,or .The set D is called the domain of the function .We call and the independent variables and the dependent variable.定义2 设二元函数的定义域为D，是D的聚点。如果存在常数A，对于任意给定的正数，总存在正数，使得当点，都有成立，那么就称常数A为函数当时的极限，记作或，也记作或。Definition 2 Let D be the domain of the function of two variables, be a point of accumulation of D.If there exists a constant A, such that, for each there is a corresponding such that ,provided that, then we call the constant A the limit of as .定义3 设二元函数的定义域为D，是D的聚点，且。如果，则称函数在点连续。Definition 3 Let D be the domain of the function of two variables, be a point of accumulation of D and . If ,then we say that is continuous at the point .定义4 设二元函数的定义域为D，是D的聚点。如果函数在点不连续，则称为函数的间断点。Definition 4 Let D be the domain of the function of two variables , be a point of accumulation of D. If is not continuous at , then we say that is a discontinuity point of .性质1（有界性与最大值最小值定理）在有界闭区域D上的多元留恋许函数，必定在D上有界且一定能取得它的最大值和最小值。Property 1 (Boundedness and max-min theorem) Let be a continuous function of several variables on a closed region D, then has an absolute maximum and an absolute minimum on the region. In particular, must be bounded on the region.性质2 （介值定理）在有界闭区域D上的多元连续函数，必定在D上取得界于最大值M与最小值m之间的任何值。Property 2 (Intermediate Value Theorem) Let be a continuous function of several variables on a closed region D, then can obtain any number between its absolute maximum M and its absolute minimum m.8.2偏导数（Partial Derivative）定义 设函数是关于的二元函数，如果固定为常数，则是关于的一元函数。它在的导数叫做在关于的偏导数，记作。即类似地，在关于的偏导数，记作，由所给出。Definition Suppose that is a function of two variables and . If is held constant , say , then is a function of the single variable .Its derivative at is called the partial derivative of with respect to at and is denoted by . Thus .Similarly, the partial derivative of with respect to at ,and is denoted by and is given by .定理 若和在开集S连续，则对于S的每一点有。Theorem If and are continuous on an open set S, then at each point of S.8.3全微分（Total Differential）定义 如果函数在点的全增量可表示为，其中，则称函数在点可微分，而称为函数在点的全微分，记作，即。Definition If the total increment of the function at can be expressed as ,where , then is said to be differentiable at . is called the total differential of at, denoted by , i.e. .定理1（必要条件）如果函数在点可微分，则该函数在点的偏导数必定存在，且函数在点的全微分为。Theorem 1 (Necessary Condition) Suppose the function is differentiable at,then its partial derivative at exist, and the total differential of at is given by . 定理2 （充分条件）如果函数在点的偏导数连续，则函数在该点可微分。Theorem 2 (Sufficient Condition) If has continuous partial derivatives at , then is differentiable at .8.4链式法则（The Chain Rule）定理1 设函数和在点t可微，在点可微。则在点t可微且。Theorem 1 Let and be differentiable at t, and let be differentiable at . Then is differentiable at t and.定理2 设在点具有一阶偏导数，且在点可微，则复合函数在点的一阶偏导数存在，且有。Theorem 2 Let and have first partial derivatives at ,and letbe differentiable at . Then has first partial derivatives at , and .8.5隐函数的求导公式（ Derivative Formula for Implicit Function）假设方程确定了y为x的隐函数，比如但是的显式表达式很难或者不可能求出来。我们依然可以求出。应用链式法则，方程两边对x求导。我们得到，解得。Suppose that defines y implicitly as a function of x, for example, but that the function is difficult or impossible to determine. We can still find . Lets differerntiate both sides of with respect to x using the Chain Rule. We obtain . Solving for.假设z是由方程所确定的x和y的隐函数，两边关于x求偏导数，保持y不变，则。注意到，即可解得，得到下面的第一个公式。类似地，固定x，对y求偏导数即得第二个公式。If z is an implicit function of x and y defined by the equation , then differentiation of both sides with respect to x, holding y fixed, yields . If we solve for and note that ,we get the first of the formulas below. A similar calculation holding x fixed and differentiating with respect to y produces the second formula.8.6多元函数微分学的几何应用（Geometric Applications of Differentiation of Functions of Several Variables）假设曲线L由参数方程所给出，且x，y，z在区间上可微。假设对应到参数值，则L在点M的切线方程为。Assume that the curve L is given by the parametric equations,and functions x,y,z are differentiable on the interval.Assume also that point corresponds the value of parameter t, then the equation of the tangent line to the curve L at point M is given by.假设是方程所确定的曲面S上的一点，F的偏导数在该点连续且不同时为零，则曲面S在点M的切平面的方程是曲面S在点M的法线方程是Suppose that is a point on a surface S with equation F(x, y, z)=O and that the partial derivativesand are continuous and not all equal to zero at M, then the tangent plane to surface S at point M is given by and the normal line to surface S at point M is given by8.7 方向导数与梯度 (Directional Derivatives and Gradients)定理 如果函数在点可微分 , 那么函数在该点沿任一方向的方向导数存在 , 且有 其中是方向 的方向余弦。Theorem Let be differentiable at. Then has a