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d mv , where m is the mass of an object and v is its velocity. dt When F has the representation F r , it is said to be conservative. The previous theorems tell us that the integrals of conservative elds of force are independent of path. Furthermore, showing that r F 0 is the preferred way of showing that F is conservative, since it involves di erentiation, while demonstrating that exists such that F r requires integration. EXAMPLE. Newton s second law for forces is F

Fig. 2-3

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An array of doubles specifying a list of 2-D vertex points in WCS coordinates. It s a simple array with a single dimension composed of alternating X and Y values (i.e., p1x, p1y, p2x, p2y, etc.). Because you must supply at least two points to create a polyline, this array must have a minimum of four elements.

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SURFACE INTEGRALS Our previous double integrals have been related to a very special surface, the plane. Now we consider other surfaces, yet, the approach is quite similar. Surfaces can be viewed intrinsically, i.e., as non-Euclidean spaces; however, we do not do that. Rather, the surface is thought of as embedded in a three-dimensional Euclidean space and expressed through a two-parameter vector representation: x r v1 ; v2 While the purpose of the vector representation is to be general (that is, interpretable through any allowable three-space coordinate system), it is convenient to initially think in terms of rectangular Cartesian coordinates; therefore, assume r xi yj zk and that there is a parametric representation x r v1 ; v2 ; y r v1 ; v2 ; z r v1 ; v2 11

Fig. 2-4

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The functions are assumed to be continuously di erentiable. The parameter curves v2 const and v1 const establish a coordinate system on the surface (just as y const, and x const form such a system in the plane). The key to establishing the surface integral of a function is the di erential element of surface area. (For the plane that element is dA dx; dy.) At any point, P, of the surface dx @r @r dv dv @v1 1 @v2 2

RULE 3 (Product Rule).

spans the tangent plane to the surface. In particular, the directions of the coordinate curves v2 const @r @r and v1 const are designated by dx1 dv and dx2 dv , respectively (see Fig. 10-3). @v1 1 @v2 2 The cross product dx1 x dx2 @r @r dv dv @v1 @v2 1 2

Dx ( f (x) g(x)) = f (x) Dx g(x) + g(x) Dx f (x)

Note Lightweight polylines don t store Z-axis or elevation information with each vertex point. Instead, the

@r @r is the area of a di erential coordinate is normal to the tangent plane at P, and its magnitude @v @v2 1 parallelogram.

(a) Dx (x 4 ) = Dx (x 3 x) algebra ua ub = ua+b = x 3 Dx (x) + x Dx (x 3 ) = x 3 (1) + x(3x 2 ) = x 3 + 3x 3 = 4x 3 (b) Dx (x 5 ) = Dx (x 4 x) = x 4 Dx (x) + x Dx (x 4 ) = x 4 (1) + x(4x 3 ) = x 4 + 4x 4 = 5x 4 (c) Dx ((x 3 + x)(x 2 x + 2)) = (x 3 + x) Dx (x 2 x + 2) + (x 2 x + 2) Dx (x 3 + x) = (x 3 + x)(2x 1) + (x 2 x + 2)(3x 2 + 1) [by the product rule] [by example (a)] and ua = ua b ub

Fig. 10-3

[by the product rule]

(This is the usual geometric interpretation of the cross product abstracted to the di erential level.) This strongly suggests the following de nition: De nition.

The reader may have noticed a pattern in the derivatives of the powers of x: Dx (x) = 1 = 1 x 0 Dx (x 2 ) = 2x Dx (x 3 ) = 3x 2 Dx (x 4 ) = 4x 3 Dx (x 5 ) = 5x 4

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