Verification of Vector Calculus Identities in Different Coordinate Systems

A vector field, such as its expansion or compression (divergence) and its rotation (curl), are intrinsic to the field itself and don't depend on the coordinate system used to describe them. Even though the formulas for divergence and curl look vastly different in Cartesian, cylindrical, and spherical coordinates, applying them to the same vector field will always give the same result.

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✍️Mathematical Proof

$\gg$Mathematical Structures Underlying Physical Laws

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Consistency of Vector Identities

The core takeaway is that the fundamental properties of vector fields, such as their divergence (source/sink) and curl (rotation), are independent of the coordinate system used to describe them. Even though the mathematical expressions for divergence and curl are different in Cartesian, cylindrical, and spherical coordinates, applying them to the same vector field (in this case, the position vector $x$ ) consistently yields the same results: $\nabla \cdot x=3$ and $\nabla \times x=0$.

Geometric Interpretation

The analysis reinforces the geometric meaning of divergence and curl.

Divergence: The position vector $x$ points radially outward from the origin. The fact that its divergence is a constant positive value (3) indicates that it acts as a uniform source everywhere. This makes sense-if you imagine a fluid flowing radially from the origin, its outward flow is consistent at every point.
Curl: The curl of $x$ is zero everywhere, which signifies that the field has no rotational component. The position vector field is purely radial; it has no tendency to "circulate" around any point.

Power of Vector Calculus

The problem serves as a practical exercise in applying the coordinate-specific formulas for vector operators. Successfully performing the calculation in all three systems and confirming the consistency of the results demonstrates the robustness and elegance of vector calculus as a tool for describing physical phenomena.

🎬Demonstration

It demonstrates that the numerical results for the divergence and curl of the position vector are coordinate-independent. The animation's pulsing vectors show the outward flow and lack of rotation, and the changing text explicitly links this visual behavior to the consistent mathematical results across all three coordinate systems.

the analysis of the divergence and curl of the position vector