Before I begin with my actual question, I think it will be helpful to provide some context. My multivariable calculus problem set has two related problems, the first of which involves a curve in a 2D vector field, and the second of which involves a curve in a 3D vector field.
The first question, which I answered correctly (according to my professor), was the following:
Given a 2D vector field $\text{Field}(x, y) = (m(x, y), n(x, y))$ and the property that the curl of the vector field is 0, agree or disagree with the statement that $\displaystyle \oint_C m(x, y) dx + n(x, y) dy \neq 0$. Additionally, the vector field contains singularities, but the curve C does not pass through any singularity.
Here, I agreed with the statement based on the fact that the curve may not PASS through any singularities, but it may still ENCLOSE the singularities, thus affecting the value of the line integral.
The next related question is very similar.
Given a 3D vector field $Field(x, y, z) = (m(x, y, z), n(x, y, z), p(x, y, z))$ and the property that the curl of the vector field is {0, 0, 0}, agree or disagree with the statement that $\displaystyle \oint_C m(x, y, z) dx + n(x, y, z) dy + p(x, y, z) dz\neq 0$. Additionally, the vector field contains singularities, but the curve C does not pass through any singularity.
For this problem, I correctly answered that the statement was false. My justification was that in 3D space, it is not possible for a curve to enclose a singularity (this can only be done with a surface, not a curve.) Because singularities only affect line integrals if they are enclosed, I stated that the singularities do not affect the line integral due to the fact that the curve neither PASSES THROUGH nor ENCLOSES any singularities.
However, my professor told me that this explanation was not enough and that I needed to invoke Stokes' theorem.
My understanding of Stokes' theorem is that it equates the line integral $\displaystyle \oint_C m(x, y, z) dx + n(x, y, z) dy + p(x, y, z) dz$ to the surface integral $\displaystyle \iint_R \text{curl F} \cdot d \vec{S}$. However, I do not understand how I would apply it here. Wouldn't singularities, by definition, prevent Stokes' theorem from correctly calculating the path integral? By Stokes' theorem, my path integral would calculate to 0 (since the curl of the vector field is 0). But is this calculation correct? And if so, what would be the best way of explaining the phenomenon?
One note: My textbook uses line integrals in the form $\displaystyle \oint_C m(x, y) dx + n(x, y) dy$ and $\displaystyle \oint_C m(x, y, z) dx + n(x, y, z) dy + p(x, y, z) dz$ to represent $\displaystyle \oint_C \vec{F} \cdot d\vec{R}$.