Integral and differentiation are two mathematical operations in modern calculus and analysis which have been commonly applied in many fields of science. Integration and differentiation are associated and linked as inverse operation by the fundamental theorem of calculus. Both integral and differentiation are defined based on the concept of additive Lebesgue measure although various generations have been developed with different forms and notations. Fractals can be considered as geometry with fractal dimension (e.g., non-integer) which no longer possesses Lebesgue additive property. Accordingly, the ordinary integral and differentiation operations are no longer applicable to the fractal geometry with singularity. This paper introduces a recently developed concept of fractal differentiation and integral operations. These operations are expressed using the similar notations of the ordinary operations except the measures are defined in fractal space or measures with fractal dimension. The calculus operations can be used to describe the new concept of fractal density, the density with fractal dimension or density of matter with fractal dimension. The concept and methods are also applied to interpret the Bouguer anomaly over the mid-ocean ridges. The results show that the Bouguer gravity anomaly depicts singularity over the mid-ocean ridges. The development of new calculus operations can significantly improve the accuracy of geodynamic models.

积分和微分是现代微积分和分析中的两个数学运算，它们已广泛应用于许多科学领域。微积分的基本定理将积分和微分联系起来并作为逆运算链接。积分和微分都是基于累加Lebesgue测度的概念来定义的，尽管以不同的形式和符号开发了不同的代。分形可以被视为具有分形维数（例如，非整数）的几何形状，它不再具有Lebesgue加性。因此，普通的积分和微分运算不再适用于具有奇异性的分形几何。本文介绍了最近发展的分形微分和积分运算的概念。除了在分形空间中定义度量或具有分形维数的度量外，这些操作使用与普通操作类似的表示法表示。微积分运算可用于描述分形密度，具有分形维数的密度或具有分形维数的物质密度的新概念。该概念和方法也用于解释洋中脊上的布格异常。结果表明，布格重力异常描述了中海脊上的奇异性。新的微积分运算的开发可以显着提高地球动力学模型的准确性。具有分形维数的密度或具有分形维数的物质的密度。该概念和方法也适用于解释洋中脊上的布格异常。结果表明，布格重力异常描述了中海脊上的奇异性。新的微积分运算的开发可以显着提高地球动力学模型的准确性。具有分形维数的密度或具有分形维数的物质的密度。该概念和方法也用于解释洋中脊上的布格异常。结果表明，布格重力异常描述了中海脊上的奇异性。新的微积分运算的开发可以显着提高地球动力学模型的准确性。