In Griffiths' Electrodynamics, there is a section in Appendix A where he sketches a proof of Stokes's theorem.

Consider a vector function $\mathbf{A}=A_u\mathbf{\hat{u}}+A_v\mathbf{\hat{v}}+A_w\mathbf{\hat{w}}$

Consider the following depiction of a point $(u,v,w)$ on a plane parallel to the uv-plane (ie $w$ is constant), and three other points obtained by way of infinitesimal displacements from $(u,v,w)$ (4th Ed, page 580):

Griffiths' Electrodynamics, Appendix A, page 580, Curl

The goal is to obtain curl in curvilinear coordinates starting from the line integral $\oint \vec{A} \cdot d\vec{l}$ "around the infinitesimal loop generated by starting at $(u,v,w)$ and successively increasing $u$ and $v$ by infinitesimal amounts, holding $w$ constant. The surface is a rectangle (at least, in the infinitesimal limit)..."

He then proceeds to actually compute each piece of the line integral. For the case of the line integral associated with the picture above, the $\hat{w}$ component of the curl is obtained.

My question is: is the depiction accurate? There are arrows going counterclockwise around the curved rectangle. But isn't the actual line integral on an actual rectangle, as in the red rectangle in the picture I drew below?

enter image description here


1 Answer 1


The red parallelogram will coincide with the green one in the infinitesimal limit.


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