The Gradient, the Del Operator, Covariant Differentiation, and the Divergence Theorem

Abstract

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Suppose that you had a topographical map of a piece of land and wanted to indicate at a spot P on the map the slope m of the land in a direction t. This could be done by drawing a vector mt from P, as indicated in Fig. 4.1. Obviously, if the terrain is smooth but not level, there is one direction from P in which the slope is a maximum. This is called the direction of steepest ascent.¹ The associated vector is called the gradient of the elevation at P. If you draw a contour line through P, you will realize that the gradient at P must be ⊥ to this contour.