Vector Field Analysis in Cylindrical Coordinates

A vector field with zero curl can still have a non-zero circulation integral if the integration path encloses a singularity, which is a point where the field is undefined. This demonstrates a crucial exception to Stokes' Theorem, which assumes the absence of such singularities within the surface of integration.

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✍️Mathematical Proof

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Divergence and Curl

The vector field $v=\frac{1}{\rho} e_\phi$ has both a divergence of zero and a curl of zero for all $\rho>0$. This means the field is "source-free" and "irrotational" in this region.

Non-zero Circulation Integral

Despite having a curl of zero, the line integral of the vector field around a closed loop is non-zero ( $4 \pi$ ). This seemingly contradictory result is because the vector field has a singularity at $\rho=0$, which is enclosed by the path of integration.

Stokes' Theorem

The analysis demonstrates a key exception to Stokes' Theorem. The theorem, which states that the circulation of a vector field is equal to the flux of its curl through a surface, only applies to surfaces that do not contain singularities of the vector field. The non-zero result of the integral, despite the zero curl, highlights the presence of a singularity at the origin.

Simplified Integration

The direct calculation of the line integral was simplified by using the property of orthogonal basis vectors, which allowed the dot product $v \cdot d x$ to be easily reduced to just $d \phi$. This streamlined the process of solving the integral along the given path.

🎬Demonstration

A vector field can have a non-zero circulation integral around a closed loop, even if its curl is zero everywhere along the path of the loop. This happens when the loop encloses a singularity—a point where the vector field is undefined. In this specific example, the field's divergence and curl are both zero everywhere except at the origin, which is the singularity. The non-zero circulation integral is a direct consequence of the path enclosing this singularity. This concept is a fundamental part of the generalized Stokes' Theorem in vector calculus.

the vector field having a curl of zero everywhere except at the origin