Similar to the gravitational curvatures in the gravity field, the magnetic curvatures (i.e., third-order derivatives of the magnetic potential) have been recently proposed in the magnetic field. The components of the magnetic curvatures are all involved in the expressions for the first-order derivatives of invariants of the magnetic gradient tensor (MGT), whose physical meaning is the change rate of invariants of the MGT and their theoretical models have not yet been fully developed. In this contribution, the general expressions for the First-order derivatives of Principal and Main Invariants of the Magnetic Gradient Tensor (i.e., FPIMGT and FMIMGT) with combined components of the MGT and magnetic curvatures are presented. Specifically, the expressions for the FPIMGT and FMIMGT of a uniformly magnetized tesseroid and spherical shell are derived as examples for basic mass bodies in the spherical coordinate system in the spatial domain for the magnetic field modeling. The near zone and polar singularity problems are numerically investigated for these newly derived expressions. Numerical experiments show that the near zone problem has been found for the FPIMGT and principal invariants of the magnetic gradient tensor (PIMGT), whereas the polar singularity problem does not occur for the FPIMGT and PIMGT when using the Cartesian integral kernels for different heights and grid resolutions. This study shows that the calculation strategy by substituting the calculated values of the MGT and magnetic curvatures into the general formulae of the PIMGT and FPIMGT can provide proper numerical precision measured by relative approximation errors dependent on the computation point’s height and latitude for the FPIMGT and PIMGT. For instance, using the grid size of \(1^{\circ }\times 1^{\circ }\) at a satellite height of 260 km, relative approximation errors in \(Log_{10}\) scale have been reduced at a level lower than − 1 for the evaluation of the FPIMGT and PIMGT. The first-order derivatives of principal and main invariants of the MGT will be applied in related magnetic field studies (e.g., magnetic detection, inversion, location, position, characterization, navigation, and exploration) to present additional information (i.e., more detailed geophysical features in terms of change rates) compared to the principal and main invariants of the magnetic gradient tensor.

与重力场中的重力曲率类似，最近在磁场中提出了磁曲率（即磁势的三阶导数）。磁曲率的分量都涉及到磁梯度张量（MGT）不变量的一阶导数的表达式，其物理意义是MGT不变量的变化率，其理论模型尚未完全发达。在这篇文章中，给出了磁梯度张量（即，FPIMGT 和 FMIMGT）的主不变量和主不变量的一阶导数的一般表达式，其中包含 MGT 和磁曲率的组合分量。具体来说，作为磁场建模的空间域中球坐标系中的基本质量体的示例，导出了均匀磁化的tesseroid和球壳的FPIMGT和FMIMGT表达式。对于这些新导出的表达式，对近区和极地奇点问题进行了数值研究。数值实验表明，FPIMGT和磁梯度张量的主不变量（PIMGT）存在近区问题，而在不同高度和网格使用笛卡尔积分核时，FPIMGT和PIMGT不会出现极点奇点问题决议。本研究表明，通过将 MGT 和磁曲率的计算值代入 PIMGT 和 FPIMGT 的一般公式的计算策略可以提供适当的数值精度，该精度由取决于计算点的高度和纬度的相对近似误差测量，用于 FPIMGT 和 PIMGT . 例如，使用网格大小\(1^{\circ }\times 1^{\circ }\)在 260 km 的卫星高度，\(Log_{10}\)尺度的相对近似误差已降低到低于 − 1 的水平FPIMGT 和 PIMGT 的评估。MGT 的主不变量和主不变量的一阶导数将应用于相关的磁场研究（例如，磁探测、反演、定位、定位、表征、导航和勘探）以提供附加信息（即更详细的地球物理变化率方面的特征）与磁梯度张量的主不变量和主不变量相比。