In recent years, the gravitational curvatures, the third-order derivatives of the gravitational potential (GP), of a tesseroid have been introduced in the context of gravity field modeling. Analogous to the gravity field, magnetic field modeling can be expanded by magnetic curvatures (MC), the third-order derivatives of the magnetic potential (MP), which are the change rates of the magnetic gradient tensor (MGT). Exploiting Poisson’s relations between (n+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n+1)$$\end{document}th-order derivatives of the GP and nth-order derivatives of the MP, this paper derives expressions for the MC of a uniformly magnetized tesseroid using the fourth-order derivatives of the GP of a uniform tesseroid expressed in terms of the Cartesian kernel functions. Based on the magnetic effects of a uniform spherical shell, all expressions for the MP, magnetic vector (MV), MGT and MC of tesseroids have been examined for numerical problems due to singularity of the respective integral kernels (i.e., near zone and polar singularity problems). For this, the closed analytical expressions for the MP, MV, MGT and MC of the uniform spherical shell have been provided and used to generate singularity-free reference values. Varying both height and latitude of the computation point, it is found numerically that the near zone problem also exists for all magnetic quantities (i.e., MP, MV, MGT and MC). The numerical tests also reveal that the polar singularity problems do not occur for the magnetic quantity as a result of the use of Cartesian as opposed to spherical integral kernels. This demonstrates that the magnetic quantity including the newly derived MC ‘inherit’ the same numerical properties as the corresponding gravitational functional. Possible future applications (e.g., geophysical information) of the MC formulas of a uniformly magnetized tesseroid could be improved modeling of the Earth’s magnetic field by dedicated satellite missions.

中文翻译：

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使用笛卡尔核的均匀磁化 Tesseroid 的磁曲率

近年来，已经在重力场建模的背景下引入了重力曲率，即重力势（GP）的三阶导数。与重力场类似，磁场建模可以通过磁曲率 (MC) 进行扩展，磁曲率 (MC) 是磁势 (MP) 的三阶导数，即磁梯度张量 (MGT) 的变化率。利用泊松关系 (n+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{ upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n+1)$$\end{document} GP 的一阶导数和 MP 的 n 阶导数，本文使用以笛卡尔核函数表示的均匀 tesseroid 的 GP 的四阶导数推导了均匀磁化 tesseroid 的 MC 表达式。基于均匀球壳的磁效应，已针对由于各自积分核的奇异性（即近区和极点奇异性）的数值问题检查了苔藓体的 MP、磁矢量 (MV)、MGT 和 MC 的所有表达式问题）。为此，已经提供了均匀球壳的 MP、MV、MGT 和 MC 的封闭解析表达式，并用于生成无奇点参考值。改变计算点的高度和纬度，从数值上发现，所有磁量（即 MP、MV、MGT 和 MC）也存在近区问题。数值测试还表明，由于使用笛卡尔而不是球积分核，磁量不会出现极点奇点问题。这表明包括新导出的 MC 在内的磁量“继承”了与相应的引力泛函相同的数值特性。均匀磁化的 tesseroid 的 MC 公式未来可能的应用（例如，地球物理信息）可以通过专门的卫星任务改进地球磁场的建模。这表明包括新导出的 MC 在内的磁量“继承”了与相应的引力泛函相同的数值特性。均匀磁化的 tesseroid 的 MC 公式未来可能的应用（例如，地球物理信息）可以通过专门的卫星任务改进地球磁场的建模。这表明包括新导出的 MC 在内的磁量“继承”了与相应的引力泛函相同的数值特性。均匀磁化的 tesseroid 的 MC 公式未来可能的应用（例如，地球物理信息）可以通过专门的卫星任务改进地球磁场的建模。