MDcalc_Diff_5_6_7 - Multidimensional Calculus Differential...

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Multidimensional Calculus: Differential Theory Chapter & Page: 9–19 9.5 General Formulas for the Divergence and Curl Later, we will discover the geometric significance of the divergence and the curl of a vector field. Then, using those, we can both redefine divergence and curl in a coordinate-free manner, and obtain more general formulas for these. This will come from our development of the divergence (or Gauss’s) theorem and the Stokes’ theorem. In the meantime, I will simply tell you what those formulas are, and we will use them, as needed, with the understanding that, eventually, we will see how to derive them. As usual, let braceleftbig( x 1 , x 2 ,..., x N )bracerightbig be an orthogonal coordinate system for our space S with { h 1 , h 2 ,..., h N } , { ε 1 , ε 2 ,..., ε N } and { e 1 , e 2 ,..., e N } being the associated scaling factors, tangent vectors and normalized tangent vectors at each point. Assume F is a vector field with component/coordinate formula F ( x ) = N summationdisplay i = 1 F i ( x 1 , x 2 ,..., x N ) e i where x ( x 1 , x 2 ,..., x N ) . The corresponding formulas for the divergence of F depend somewhat on the dimension N of the space, though the pattern will become obvious. If N = 2 , then · F = 1 h 1 h 2 braceleftbigg x 1 bracketleftbig F 1 h 2 bracketrightbig + x 2 bracketleftbig F 2 h 1 bracketrightbig bracerightbigg . (9.16a) If N = 3 , then · F = 1 h 1 h 2 h 3 braceleftbigg x 1 bracketleftbig F 1 h 2 h 3 bracketrightbig + x 2 bracketleftbig F 2 h 1 h 3 bracketrightbig + x 3 bracketleftbig F 3 h 1 h 2 bracketrightbig bracerightbigg . (9.16b) If N = 4 , then · F = 1 h 1 h 2 h 3 h 4 braceleftbigg x 1 bracketleftbig F 1 h 2 h 3 h 4 bracketrightbig + x 2 bracketleftbig F 2 h 1 h 3 h 4 bracketrightbig + x 3 bracketleftbig F 3 h 1 h 2 h 4 bracketrightbig + x 4 bracketleftbig F 4 h 1 h 2 h 3 bracketrightbig bracerightbigg . (9.16c) And so on. The curl still only makes sense if the dimension N is three. Using Stokes’ theorem, we will version: 10/31/2011
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Multidimensional Calculus: Differential Theory Chapter & Page: 9–20 be able to show that × F = 1 h 1 h 2 h 3 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle h 1 e 1 h 2 e 2 h 3 e 3 x 1 x 1 x 1 h 1 F 1 h 2 F 2 h 3 F 3 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = 1 h 2 h 3 parenleftbigg x 2 bracketleftbig h 3 F 3 bracketrightbig x 3 bracketleftbig h 2 F 2 bracketrightbig parenrightbigg e 1 1 h 1 h 3 parenleftbigg x 1 bracketleftbig h 3 F 3 bracketrightbig x 3 bracketleftbig h 1 F 1 bracketrightbig parenrightbigg e 2 + 1 h 1 h 2 parenleftbigg x 1 bracketleftbig h 2 F 2 bracketrightbig x 2 bracketleftbig h 1 F 1 bracketrightbig parenrightbigg e 3 .
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