也就是二重和三重积分的内容，以及平面和空间中的向量积分。
And so that was stuff about double and triple integrals and vector calculus in the plane and in space.
这个曲面积分，不是同一个向量场。
OK, and that surface integral, well, it's not for the same vector field.
一个向量场,如果定义在单连通区域并且旋度为零，那么它就是一个梯度场，并且其上的线积分与路径无关。
OK, so, we've seen that if we have a vector field defined in a simply connected region, and its curl is zero, then it's a gradient field, and the line integral is path independent.
了解这两种表述后，我们不仅需要向量场，就是左边这里，这是曲线c上的线积分，向量场在曲线上有定义。
So, in both cases, we need the vector field to be defined not only, I mean, the left hand side makes sense if a vector field is just defined on the curve because it's just a line integral on c.
一个保守的向量场就是说,沿任意闭曲线的线积分的结果是。
So, to say that a vector field with conservative means 0 that the line integral is zero along any closed curve.
需要建立向量场点乘nds的通量积分。
So, we need to set up the flux integral for a vector field dot ndS.
我将要证明，一个只有z分量的向量场的通量,等于一个三重积分，其中被积表达式为。
I'm going to prove that the flux of a vector field that has only a z component is actually equal to the triple integral of, RzdV well, the divergence of this is just R sub z dV.
其中一种说明了，在向量场上，沿逆时针方向，向量做的功等于，平面区域上旋度F的二重积分。
So, one of them says the line integral for the work done by a vector field along a closed curve counterclockwise is equal to the double integral of a curl of a field over the enclosed region.
的法向量似乎是指向外部。，And, the,normal, vector, to,s, seems, to, be, pointing, out wards, everywhere。，如果我们有一个法向量指向外部的封闭曲面，并且我们要求出它的一个通量积分，那么我们可以用一个三重积分来代替这个通量积分。
S if we have a closed surface with a normal vector pointing outwards, and we want to find a flux integral for it, well, we can replace that with a triple integral.
这把一个向量场的线积分，和另外一个向量场的曲面积分联系起来。
This relates a line integral for one field to a surface integral from another field.
根本上说，以下性质是等价的，一个向量场是梯度场，等价于积分与路径无关，也等价于向量场是保守的。
And, basically, we say that these properties are equivalent being a gradient field or being path independent or being conservative.
其实是要对一个与其无关的向量场积分，其实是要对一个与其无关的向量场积分。
We are just integrating a vector field that has nothing to do with that.
讨论局部对称空间中具有平行平均曲率向量的子流形，得到其关于第二基本形式模长平方的积分不等式的相关定理。
This paper discusses submanifolds with parallel mean curvature vector in local symmetric Spaces and obtains integral invariants about the square of modulus-length.
课程内容包括常微分方程、空间解析几何与向量代数、多元函数微分学、多元函数积分学和无穷级数等几大板块。
This course consists of several major parts such as ordinary differential equation vectors and analytic geometry derivatives integration and series.
课程内容包括空间解析几何与向量代数、多元函数微分学、多元函数积分学和无穷级数等几大板块。
This course consists of several major parts such as vectors and analytic geometry derivatives integration and series.
在卫星的状态估计过程中应用推广的序列估计算法，借助数值积分方法积分状态向量和协方差矩阵。
For the estimation of satellite state, the extended sequential estimation algorithm was applied. The numerical method was used to integrate state vector and error covariance matrix.
你懂得向量微积分，但你不记得如何做长除法。
You know vector calculus but you can't remember how to do long division.
本文提出了一种自适应控制算法：单位向量加积分校正自适应算法。
A new control algorithm is proposed in this paper. That is the unit vector with a integral corrective term adaption law.
研究了拟常曲率流形中具有平行平均曲率向量的子流形，给出了两个积分不等式。
We study the submanifolds with parallel mean curvature vector in a manifold of quasi constant curvature, and give two integrate inequalities.
计算积分的方法、复数方法、无线级数、奇殊函数、微分方程、向量及矩阵、群论。
Methods in evaluating integrals, some complex variable methods, infinite series, special function, ordinary differential equations, vector and materials, groups and group representations.
然后，由虚功原理推导出了整体平衡方程中的广义刚度矩阵和荷载向量的具体表达式，并对其数值积分方法进行了讨论。
Then, the generalized stiffness matrix and load vector in the equilibrium equations are formulated by using the virtual work principle, and the numerical integration method is discussed.
利用平均电价差值积分模型将电力市场中发电商的机组报价曲线转换为一维特征向量，从而采用传统聚类方法对机组报价曲线实现分类。
This model can transform units bidding curve of power producer in market into a one-dimensional feature vector, so it can implement classification of units bidding using classical clustering method.
建立了半鞅向量随机积分的一个结果，能方便处理可料过程在向量随机积分意义下对半鞅的分解随可料过程不同而不同的问题。
It establishes a result which can be easily applied to the problem derived from the different predictable process in the decomposition of semimartingales in the sense of vector stochastic integrals.
由涡的定义导出的向量泊桑方程的积分形式的解析解，被用于低速风洞壁引起的干扰速度的计算。
The analytical solution of Poisson's equation, derived from the definition of vortex, has been applied to the calculations of interference velocities due to the presence of wind tunnel walls.
像是向量微积分、微分和积分形式的麦斯威尔方程序。
Some knowledge of vector calculus assumed. Maxwell's equations, in both differential and integral form.
本课程主要内容包括：向量代数与空间解析几何、多元函数微积分、无穷级数等。
This course mainly includes: vector algebra and analytic geometry in space, multivariable calculus and infinite series.
本课程主要内容包括：向量代数与空间解析几何、多元函数微积分、无穷级数等。
This course mainly includes: vector algebra and analytic geometry in space, multivariable calculus and infinite series.
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