The Vector Differential ($d\rr$)

The infinitesimal vector displacement, $d\rr$, is shown above from several points of view. On the left, $d\rr$ represents a small step along a curve; on the right, $d\rr$ is expanded in terms of both rectangular and polar coordinates, leading to the expressions \begin{equation} d\rr = dx\,\xhat + dy\,\yhat = dr\,\rhat + r\,d\phi\,\phat \label{drdef} \end{equation} The geometric notion of $d\rr$ as an infinitesimal vector displacement is a unifying theme that helps in visualizing the geometry of all of vector calculus.


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