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Unformatted text preview: 9 Multidimensional Calculus: Mainly Differential Theory In the following, we will attempt to quickly develop the basic differential theory of calculus in multidimensional spaces. You’ve probably already seen much of this theory. Hopefully, we will develop a better understanding of the material than is usually imparted in the more elementary treatments, and see how to extend it to more general spaces and coordinate systems. By the way, in keeping with the common practice in physics of denoting the position of a moving object by r , I will relent and often use this notation instead of x . 9.1 Motion, Curves, Arclength, Acceleration and the Christoffel Symbols For all the following, assume we are considering motion in some space of positions S , and that braceleftbig( x 1 , x 2 , . . . , x N )bracerightbig is any coordinate system for this space. As before, { h 1 , h 2 , . . . , h N } , { ε 1 , ε 2 , . . . , ε N } and { e 1 , e 2 , . . . , e N } are the associated scaling factors, tangent vectors and normalized tangent vectors. We also assume that we have some positionvalued function r ( t ) that traces out a curve C as t varies over some interval ( t , t 1 ) . Since this is a math/physics course, we naturally view r ( t ) as being the position of an object, say, George the Gerbil, at time t . In terms of our coordinate system, we have some coordinate formula for r r ( t ) ∼ ( x 1 ( t ), x 2 ( t ), . . . , x N ( t ) ) for t < t < t 1 . Velocity and Speed The formula for velocity v at any given time is easily computed using the chain rule: v = d r dt = N summationdisplay i = 1 ∂ r ∂ x i dx i dt = N summationdisplay i = 1 h i e i dx i dt , 10/28/2011 Multidimensional Calculus: Differential Theory Chapter & Page: 9–2 which we may prefer to rewrite as v = d r dt = N summationdisplay i = 1 h i dx i dt e i or v = d r dt = N summationdisplay i = 1 dx i dt ε i . The corresponding speed, then, is ds dt = vextenddouble vextenddouble vextenddouble vextenddouble d r dt vextenddouble vextenddouble vextenddouble vextenddouble = radicaltp radicalvertex radicalvertex radicalbt bracketleftBigg N summationdisplay i = 1 h i dx i dt e i bracketrightBigg · bracketleftBigg N summationdisplay j = 1 h j dx j dt e j bracketrightBigg . That is, ds dt = radicaltp radicalvertex radicalvertex radicalbt N summationdisplay i = 1 N summationdisplay j = 1 h i h j dx i dt dx j dt e i · e j = radicaltp radicalvertex radicalvertex radicalbt N summationdisplay i = 1 N summationdisplay j = 1 dx i dt dx j dt g i j . (9.1) If the coordinate system is orthogonal, this reduces to ds dt = radicaltp radicalvertex radicalvertex radicalbt N summationdisplay i = 1 parenleftbigg h i dx i dt parenrightbigg 2 . (9.2) !...
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 Summer '11
 Staff
 Math, Calculus, dt, Coordinate system, Spherical coordinate system, Polar coordinate system, Christoffel symbols

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