坐标变换在偏微分方程中的应用

坐标变换在偏微分方程中的应用(任务书,开题报告,论文10000字)
摘要
坐标变换作为求解不同情况的问题，已有广泛的应用。本文研究利用坐标变换考察方程的变化以及边界的变化，使得方程的形式发生了变化，尤其是边界的映射，将原有问题转化为了另一个问题，从而引出一些非常有意义和有趣的结论。
变换光学的理论就是建立在两个不同坐标系之间相互的坐标变换以及麦克斯韦方程组形式不变性的基础上而得到的，这样方便在处理实际问题时，将实际情况中较复杂的坐标系变换到相对合适的坐标系下来求解，同时也为隐形设备的设计提供了理论基础。
基于麦克斯韦方程在坐标变换下形式不变性的变换光学理论使得隐形斗篷成为了可实现的科学研究，而且通过坐标变换可以反向设计材料，于是现实中真正的隐形斗篷慢慢的被成功的研究了出来，如今隐形斗篷的理论研究已经渐渐成熟，通过超材料已经有了许多成功的实践，这项技术将是未来非常重要的一项科学研究，在很多领域尤其军事领域的应用将是一项了不起的科技技术。
本文以坐标变换在麦克斯韦方程组的应用来展示坐标变换应用的神奇效果，具体内容如下：
(1) 介绍了坐标在坐标系间的转换，展示其映射关系。 [来源：http://think58.com]
(2) 介绍了坐标变换的基础理论，将坐标变换用数学理论展示出来，方便证明计算和理论推演。
(3) 详细说明麦克斯韦方程组在坐标变换下的方程形式不变性，以及基于此理论的变换光学理论，并以此为例展示坐标变换在偏微分方程中的神奇应用。
(4) 将基于变换光学理论而产生的隐形斗篷进行理论研究，对其进行理论推演证明，本文只是进行简单介绍以展示坐标变换应用的魅力之处。

关键词：坐标变换；坐标系；麦克斯韦方程组；变换光学；隐形斗篷
 
    Abstract
Coordinate transformation as a way to solve different situations, it has been widely used. This paper will discuss the use of coordinate transformations and changes in the border inspection equation, the form of the equation has changed, especially in mapping the boundary, the original problem is converted to another problem, which leads to some very interesting and meaningful conclusions.
Optical transformation theory is based on mutual coordinate conversion between two different coordinate systems and the invariance of the form of Maxwell Equations. It will be very convenient in dealing with practical problems that the actual situation in more complex coordinate transformation is relative to the appropriate coordinate system to solve the problem, and it also provides a theoretical basis for the design of cloaking device. [来源：http://www.think58.com]
Optical transformation theorywhich is based on the invariance of the form of Maxwell Equations makes invisibility cloak became achievable scientific research, and it can be used to the reverse engineering of material, so the real invisibility cloak slowly being successful developed.Now invisibility cloak theory has gradually matured, many successful practices have been made through the material. This technology will be a very important scientific researchin the future. In many fields, especially in the military field,these applications will be a great technology.
    In this paper, coordinate transformation in the application of Maxwell Equations will show themagical effects of coordinate transformation, as follows:
(1) Introduce the coordinate transformation between the coordinate system conversion, show their mappings.
(2) Introduce the basic theory of coordinate transformation, showthe mathematicaltheoryofcoordinate transformation, it will be used to prove conclusion, computeand do theoretical deduction. [资料来源：http://think58.com]
(3) Introduce a detailed description of the invariance of the form of MaxwellEquations under coordinate transformation. Optics transformation theory is based on this theory, and as an example to show the magical effects of coordinate transformation in partial differential equations.
(4) Discuss theinvisibility cloak theorywhich is based on the theory ofoptics transformation, prove theoretical deduction. This is only a brief introduction to demonstrate the charm ofthe application of coordinate transformation.
Keywords:Coordinate transformation; Coordinate system; Maxwell Equations;Optics transformation; Invisibility cloak
 
目录
摘  要    I
Abstract    II
第1章 绪论    1
1.1 背景分析    1
1.2 研究现状    2
1.3 主要内容    2
第2章 坐标变换基础理论    3
2.1 正交曲线坐标系中梯度、散度、旋度与调和量的表达式    3 [资料来源：http://THINK58.com]
2.1.1梯度表达式    3
2.1.2散度的表示式    3
2.1.3调和量的表达式    4
2.1.4旋度的表达式    4
2.2 梯度、散度、旋度与调和量在柱面坐标系和球面坐标系中的表达式    5
2.2.1在柱面坐标系中：    5
2.2.2在球面坐标系中：    6
第3章 坐标变换在偏微分方程中的应用    7
3.1 坐标变换    7
3.2 麦克斯韦方程组    11
3.2.1 光学变换的理论基础    11
3.2.2 麦克斯韦方程组的坐标变换形式不变性，坐标变换与变换媒质：    11
3.3 参数分析    13
3.3.1 坐标变换理论以及隐身斗篷理论分析    13
3.3.2 直角坐标系下的参数张量    15
第4章 隐形斗篷的研究    19
4.1隐形斗篷    19
4.2任意截面的圆柱斗篷    20
4.3三维斗篷    22
第5章 总结与展望    23 [资料来源：www.THINK58.com]
5.1 论文总结    23
5.2 研究展望    23
参考文献    25
致谢    26 [资料来源：THINK58.com]