🧄Proving the Generalized Curl Theorem

The proof for this relies on applying the well-known Regular Curl Theorem (Stokes' Theorem) to a cleverly chosen vector field, effectively demonstrating how core principles of vector calculus can be used to derive new, elegant relationships. The visualization reinforces this by showing how gradient vectors, which point only in the direction of steepest ascent, have no rotational component, thus confirming that their curl is indeed zero.

🎬the line integral of a scalar field around a closed path is equal to the surface integral of the curl of its gradient

This demo is that the generalized curl theorem, beautifully illustrates a fundamental principle: the path integral of a scalar field around a closed loop is always zero. Since the curl of a gradient is non-existent, the surface integral of a scalar field’s gradient over any surface will always be zero, which means its boundary integral must also be zero. The demo visually reinforces this by showing that the gradient vectors radiate outward without any swirling or rotational component.

🖊️Mathematical Proof

Proving the Generalized Curl Theorem | Proof and Derivationviadean on Notion