我们说，当一个二元函数存在时，就有梯度向量。
So, we said when we have a function of two variables, we have the gradient vector.
我们已经知道，怎样通过图像去了解二元和三元函数。
We have learned how to think of functions of two or three variables in terms of plotting them.
元函数对与谱向量对通过反射阵建立映射关系。
By reflection matrix the mapping relation of element function pair and spectral vector pair can be established.
我们学过怎样认识二元函数,从它的图像和等高线的角度来看。
we've seen is how to actually view a function of two variables in terms of its graph and its contour plot.
冻土中的未冻水的化学势是未冻水含量和温度的二元函数。
As far as the frozen soil is concerned, the chemical of unfrozen is a function of the temperature and the amount of unfrozen water.
当然我们还不能看到，三元函数的图像，因为它处在一个四维空间里。
Except, of course, we haven't see the graph of a function of three variables because that would live in 4-dimensional space.
对于二元函数的二重极限，重点是极限的存在性及其求解方法。
For double limit of a binary function, the existence of the limit and the methods to seek its solution are emphasis among our focus.
友元函数用来访问类的非公有成员。友元函数不是类的成员函数。
A friend function is used for accessing the non-public members of a class.
实验测试结果表明，该算法对一元函数和多元函数都有很好的效果。
Simulation test illustrates that the presented algorithm is efficient for both one-variable functions and multi-variable functions.
以中值定理为工具，给出了利用一阶偏导数判定二元函数极值的方法。
The criterion extreme value of binary function for first partial derivative is given by means of the mean value theorem. The two theorems are obtained.
但这种办法并不普遍，只能用于二元函数，来判断是极小值点、极大值点还是鞍点。
I should say that is for a function of two variables to try to decide whether a given critical point is a minimum, a maximum or a saddle point.
并且我们已经做过两次了，一次是关于二元函数的，另一次关于三元函数的。
And we have seen them twice. We have seen them once for functions of two variables, once for functions of three variables.
讨论了二元函数中值定理中间值的渐近性质，给出了一个相关反问题的解。
The asymptotic properties of mean value in the mean value of bivariate functions are discussed, a solution is presented for related inverse problem.
经过归纳，总结，找到一些规律：即选择适当路径以确定二元函数极限不存在。
After summing up, summarizing, find some laws: Choosing proper route to confirm the function limit of two variables' doesn't exist.
本文将一元函数的罗尔定理推广到多元函数中，并给出了一个简洁、颖的证明。
In this paper, the Rolle's theorem of a variadle function is extended to multiple-variable functions. The paper, then, gives a new and simple proof.
给出二元函数型来定式的各种值法则，包括一阶极限判定法则和二阶极限判定法则。
This paper gives several determined value rules to type of indeterminate forms of functions of two variables. These contain the first order and the second order decision rules of the limit.
通过利用泰勒公式推导出判断三元函数极值存在的充分条件和极值不存在的必要条件。
By using Taylor's formula, the article deduces the sufficient condition of the existent extreme value and the necessary condition of the inexistent extreme value of trinary function.
给出了一元函数在区间上一致连续的一个等价条件，并运用它证明了一些函数的一致连续性。
AN equivalent condition of function's consistent continuity is proved. Application of it in several examples is showed.
并根据数学分析中一元函数和多元函数的极限的相关知识，在理论上证明了该指标的有效性。
According to unary function limit and multivariate function limit of mathematical Analysis, the validity of the index Vnew is proved in theory.
为了访问一个用户定义的类里的私有数据，你必须声明全局函数为友元函数，并且在类中定义。
But to be able to reach any private data within a user defined class you have to declare the global function as a friend within the definition of the class.
利用多元函数极值的定义和偏导数的定义公式证明二元函数与一元函数在某点取得极值的关系。
This article use the diverse function extremum and the local derivative's definition formula to prove the relation between extremum obtained by the two function and the dollar function at some point.
课程内容包括函数、极限与连续、一元函数微分学、一元函数积分学和常微分方程等几大板块。
This course consists of several major parts, such as Functions, Limits and Continuity, The Derivative, Integration and Ordinary Differential Equation.
通过对两类特殊三元函数极值问题的探讨，得到了由其系数来判别它们能否取得极值的充分条件。
Via the discussion about the extremum question of two kinds of special ternary function, Obtain the sufficient condition of judging if they can get the extremum by their coefficients.
本文给出一类多元函数—三元函数是否存在极值的快速判别方法，并讨论它在实际问题中的应用。
In this paper, a convenient judgement method about extreme value of one class multivariate functions-trivariate functions was given, and its application in reality was discussed.
确切的说，对于某一天的期权报价，隐含波动率是执行价格和存续期的二元函数，呈现曲面的形态。
More precisely, for one day 'quote, the implied volatility is a function of two parameters: the strike price and the time to maturity, exhibiting a surfaced shape.
探讨二元函数的连续性、偏导数、方向导数以及可微性之间的关系，有助于我们对二元函数的学习。
A discussion of the relations between continuity, partial derivative, directional derivative and differentiability of binary function is helpful for us to study binary function.
名为spacefriend的友元函数，它是space类的友元，它的参数是类对象，输出类对象的x变量的值。
A friend function named spacefriend, which is a friend of class space, and which takes the class object as an argument and prints the value of its variable x.
然后，在此框架下提出了基于遗传规划的单个神经元的设计方法，该方法可实现对神经元函数类型的优化。
Then, based on GP the design algorithm of single neuron, which realizes the auto-optimization of neuron function types, is proposed.
然后，在此框架下提出了基于遗传规划的单个神经元的设计方法，该方法可实现对神经元函数类型的优化。
Then, based on GP the design algorithm of single neuron, which realizes the auto-optimization of neuron function types, is proposed.
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