00080___31fb0c19d8fad3f7ef258f7a75f857e4 - (4), we...

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Sec. 2.15 GEOMETRIC INTERPRETATION OF COVARIANT DERIVATIVES 59 Thus the derivative of the vector v is resolved into two parts: one arising from the variation of the components va as the coordinates e1,82,e3 are changed, the other arising from the change of the base vector gi as the position of the point 02 is changed. It is shown below that, in a Euclidean space, Hence Thus, the covariant derivative waJj represents the components of dvlaej referred to the base vectors g,. To establish Eq.
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Unformatted text preview: (4), we differentiate the equation gij = gi. gj to obtain On permuting the indices i, j, k, we can obtain the derivatives agijlaej, ag,k/aOi. Furthermore, since gi = dR/dei, we have Hence, by substitution, it is easy to show that On multiplying the two sides of the equation by gak and summing over k, the left-hand side becomes I?$, according to the definition on Christoffel symbols. We then multiply the scalar quantity on both sides of the equation by the vectors g, and sum over a to obtain...
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This note was uploaded on 01/04/2012 for the course ENG 501 taught by Professor Thomson during the Fall '05 term at MIT.

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