The sources outline a mathematical and physical journey that begins with the rigorous verification of vector calculus identities before applying them to the fundamental laws of gravitation. By moving from abstract position vectors to the interior dynamics of a planet like Earth, the narrative demonstrates how mathematical consistency allows us to describe the universe with precision. The verification of vector calculus identities across different coordinate systems serves as the mathematical bedrock for understanding physical fields as intrinsic, invariant entities. By calculating the divergence and curl of a position vector $\vec{x}$ in Cartesian, cylindrical, and spherical coordinates, the sources demonstrate that the results—a divergence of 3 and a curl of 0—remain identical regardless of the chosen frame of reference. This coordinate invariance confirms that these operators are not mere mathematical artifacts but tools that reveal the fundamental nature of space, such as the uniform radial expansion of a position field or its irrotational structure. Such verification ensures that when these tools are applied to more complex phenomena, like the gravitational field, the resulting physical laws (e.g., Gauss’s Law for Gravity) are based on robust, universal principles rather than the constraints of a specific coordinate perspective.
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