MULTIPLE INTEGRALS in .NET

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MULTIPLE INTEGRALS
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In case r is not of the type shown in the above gure, it can generally be subdivided into regions r1 ; r2 ; . . . which are of this type. Then the double integral over r is found by taking the sum of the double integrals over r1 ; r2 ; . . . . TRIPLE INTEGRALS The above results are easily generalized to closed regions in three dimensions. For example, consider a function F x; y; z de ned in a closed three-dimensional region r. Subdivide the region into n subregions of volume Vk , k 1; 2; . . . ; n. Letting k ; k ; k be some point in each subregion, we form
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F k ; k ; k Vk
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where the number n of subdivisions approaches in nity in such a way that the largest linear dimension of each subregion approaches zero. If this limit exists, we denote it by F x; y; z dV 7
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called the triple integral of F x; y; z over r. The limit does exist if F ; x; y; z is continuous (or piecemeal continuous) in r. If we construct a grid consisting of planes parallel to the xy, yz, and xz planes, the region r is subdivided into subregions which are rectangular parallelepipeds. In such case we can express the triple integral over r given by (7) as an iterated integral of the form ' ! b g2 a f2 x;y b g2 x & f2 x;y F x; y; z dx dy dz F x; y; z dz dy dx 8
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(where the innermost integral is to be evaluated rst) or the sum of such integrals. The integration can also be performed in any other order to give an equivalent result. The iterated triple integral is a sequence of integrations; rst from surface portion to surface portion, then from curve segment to curve segment, and nally from point to point. (See Fig. 9-4.) Extensions to higher dimensions are also possible.
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MULTIPLE INTEGRALS
TRANSFORMATIONS OF MULTIPLE INTEGRALS In evaluating a multiple integral over a region r, it is often convenient to use coordinates other than rectangular, such as the curvilinear coordinates considered in s 6 and 7. If we let u; v be curvilinear coordinates of points in a plane, there will be a set of transformation equations x f u; v ; y g u; v mapping points x; y of the xy plane into points u; v of the uv plane. In such case the region r of the xy plane is mapped into a region r 0 of the uv plane. We then have   @ x; y   du dv  9 F x; y dx dy G u; v  @ u; v 
r r0
where G u; v  Ff f u; v ; g u; v g and  @x @x    @ x; y  @u @v      @ u; v  @y @y    @u @v
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is the Jacobian of x and y with respect to u and v (see 6). Similarly if u; v; w are curvilinear coordinates in three dimensions, there will be a set of transformation equations x f u; v; w ; y g u; v; w ; z h u; v; w and we can write    @ x; y; z   11 F x; y; z dx dy dz G u; v; w  @ u; v; w  du dv dw
r r0
where G u; v; w  Fff u; v; w ; g u; v; w ; h u; v; w g and   @x   @u  @ x; y; z  @y  @ u; v; w  @u   @z   @u
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