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The Gradient Operator, Covariant Differentiation, and the Divergence Theorem

  • James G. Simmonds
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

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 m t 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.

Keywords

Covariant Derivative Tensor Field Gradient Operator Divergence Theorem Component Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1982

Authors and Affiliations

  • James G. Simmonds
    • 1
  1. 1.Department of Applied Mathematics and Computer Science Thornton HallUniversity of VirginiaCharlottesvilleUSA

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