Newton-Leibnitz expression:
　　 shows:in the siding to siding block of the integration, it can through the origin function in this siding to siding block of the end point (boundary) of the value to express. In order to promote it, it shall start to conduct analogy: ① close regionchanges to the close regionD；② end point a,b change into the boundary L；③ function
　　 changes into；④ definite integration
　　 changes into double integration
　　and;⑤ , changes into and；⑥expression eprobably canchange into：
　　
　　In order to explore the problem⑥，consider the close region D is not only the X one pattern region, but also the Y一pattern region，namely
D=
　　Firstly , regards the Das the X一pattern region，as picture1，then
　　
　　 . Then regard Das Y one pattern region，likeness it can gain
　　. Due to that in the settle of the process of the above two expressions, they are mutual independent, therefore, the integrand doesn’t need to be the same one. According to the habit, the integrand respectively regarded as，it can gain.
　　3.1.3 To deeply explore and form the module
　　Whether the space region of the triple integration can through the curve surface integration of the along with its border∑ to express? Whether the smooth curve surface ∑ of the curvesurface integrationcan through the curve integration along with its border curve L to express? As to the two problems’ exploration, it can reach the Mr. High formula and stokes expressions.
　　Newton—Leibnitz expression, Green expression, Mr. High formula and stokes expressions all respectively give the relationship of the integration in certain region and the integration along this region along its border. Whether it can use the uniform way to describe these formulas and relation of the implication differential of these formulas and integration?Furthermore, can these formulas be promoted to the high dimension space? Through introducing the outer product and differential form and define the differential form of the differential and integration, it can discover the above four formulas can reflect in the one dimension, two dimension and three dimension of the K times differential formof the outer differential in the k+1 dimension region△ of the integration is equal toin the K dimension △ of the integration, to promote this conclusion to the n dimension space, it can reach the general stokes expression:
　　
　　Here, the k is the nature number that is smaller than n.△is the k+1 dimension region in n dimension space,△is the k dimension region in n dimension space. Therefore, the general stokes expression disclose theundoing effect of the outer differential calculation and the integration[7]，which embody theinverse relation between high dimension space of the differential and the integration.
　　3.2 To explore the application of the Green formula
　　3.2.1 The direct application of the Green formula
　　The direct application of the Green formula is mainly embodying in one that it is making use of the curve integration to calculate the double integration; two it is to make use of the double integration to calculate the curve integration.
　　3.2.2 The further exploration of the curve integration calculation
　　（1）in the enlighten of the two examples[7]——to search conception from the studying experience
　　In the first example, to make A as the starting point and to make B as the ending point of the curve integration, the integration values is different as along with different routes. However in the example2, the curve integration value only has relation with the ending point but has nothing to do with the selection of the routes.
　Here, “that it has nothing to do with the route “has attractive to use.When the curve integration goes along with different route, and the value is equal, then the arbitrarily selected route shall be equal. Therefore, it can gain that the definition that plane curve integration has nothing to do with the routes.
　　(2) to produce method from the conception
　　The definition that the plane of the curve integration has nothing to do with the route is rather delicate. However, it is useless-----lacking feasibility. Therefore, it shall combine the definition with the previous learning content so as to find the convenient method.
　　
　　As picture 2, G is a region, A、B isthe arbitrarily destine of the two points in G，L1、L2为is the arbitrary two curve from A to B in G. Because in在G has nothing to do with the route.
　　Therefore
　　Thenandcan enclosea arbitrary close curve cin G c，therefore there is：
　　What is the function of this conclusion? Because c is an arbitrary close curve in G, therefore, the conclusion judgment is still not convenient---it can not be calculated the results. However, this conclusion can make people think about other door----Green formula: (责任编辑：南粤论文中心)转贴于南粤论文中心: http://www.nylw.net(南粤论文中心__代写代发论文_毕业论文带写_广州职称论文代发_广州论文网）