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@r @z j k @v2 @y in .NET
@r @z j k @v2 @y Recognize QR Code ISO/IEC18004 In .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in .NET framework applications. Denso QR Bar Code Drawer In .NET Using Barcode generation for .NET framework Control to generate, create Quick Response Code image in Visual Studio .NET applications. 14 QR Code JIS X 0510 Reader In .NET Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications. Drawing Barcode In .NET Using Barcode creator for VS .NET Control to generate, create barcode image in .NET applications. If the surface is given in the implicit form F x; y; z 0, then the gradient may be employed to obtain another representation. To establish it, recall that at any surface point P the gradient, rF is perpendicular (normal) to the tangent plane (and hence to S). Therefore, the following equality of the unit vectors holds (up to sign): Bar Code Recognizer In VS .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in Visual Studio .NET applications. Make QR Code ISO/IEC18004 In C#.NET Using Barcode encoder for .NET framework Control to generate, create Denso QR Bar Code image in .NET applications. CHAP. 10] Paint QRCode In Visual Studio .NET Using Barcode creation for ASP.NET Control to generate, create Denso QR Bar Code image in ASP.NET applications. Create QR Code 2d Barcode In VB.NET Using Barcode creation for Visual Studio .NET Control to generate, create QR Code 2d barcode image in .NET applications. LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS
Generate 2D Barcode In Visual Studio .NET Using Barcode maker for Visual Studio .NET Control to generate, create 2D Barcode image in Visual Studio .NET applications. Barcode Generation In .NET Using Barcode generator for .NET Control to generate, create bar code image in .NET framework applications. 0 @r rF @r @r @r @v jrFj @x @y @v2 1
Printing Barcode In .NET Using Barcode creator for .NET framework Control to generate, create bar code image in Visual Studio .NET applications. UPCE Supplement 2 Generator In Visual Studio .NET Using Barcode creation for .NET Control to generate, create UPC  E0 image in VS .NET applications. 15 Paint ECC200 In None Using Barcode creator for Font Control to generate, create DataMatrix image in Font applications. UPCA Creation In None Using Barcode encoder for Font Control to generate, create UPCA image in Font applications. [Now a conclusion of the theory of implicit functions is that from F x; y; z 0 (and under appropriate conditions) there can be produced an explicit representation z f x; y of a portion of the surface. This is an existence statement. The theorem does not say that this representation can be explicitly produced.] With this fact in hand, we again let v1 x; v2 y; z f v1 ; v2 . Then rF Fx i fy j Fz k Taking the dot product of both sides of (15) yields Fz 1 @r @r jrFj @v @v2 1 The ambiguity of sign can be eliminated by taking the absolute value of both sides of the equation. Then 2 2 2 1=2 @r @r jrFj Fx Fy Fz @v @v2 jFz j jFz j 1 and the surface integral of takes the form Fx 2 Fy 2 Fz 2 1=2 dx dy jFz j Generating UPCA Supplement 2 In VB.NET Using Barcode creation for VS .NET Control to generate, create UPC Symbol image in .NET framework applications. Making Code 128 Code Set B In VB.NET Using Barcode printer for .NET framework Control to generate, create Code 128A image in .NET framework applications. 16 Barcode Creation In .NET Using Barcode creation for ASP.NET Control to generate, create bar code image in ASP.NET applications. Generating Bar Code In ObjectiveC Using Barcode drawer for iPhone Control to generate, create barcode image in iPhone applications. The formulas (14) and (16) also can be introduced in the following nonvectorial manner. Let S be a twosided surface having projection r on the xy plane as in the adjoining Fig. 104. Assume that an equation for S is z f x; y , where f is singlevalued and continous for all x and y in r . Divide r into n subregions of area Ap ; p 1; 2; . . . ; n, and erect a vertical column on each of these subregions to intersect S in an area Sp . Decode UPC A In C# Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications. ANSI/AIM Code 39 Scanner In Visual Basic .NET Using Barcode reader for .NET Control to read, scan read, scan image in Visual Studio .NET applications. Fig. 104 Let x; y; z be singlevalued and continuous at all points of S. Form the sum
n X p 1
p ; p ; p Sp
17 LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS
[CHAP. 10
where p ; p ; p is some point of Sp . If the limit of this sum as n ! 1 in such a way that each Sp ! 0 exists, the resulting limit is called the surface integral of x; y; z over S and is designated by x; y; z dS 18 Since Sp j sec
p j Ap approximately, where
p is the angle between the normal line to S and the positive zaxis, the limit of the sum (17) can be written x; y; z j sec j dA 19
The quantity j sec
j is given by 1 j sec
j jnp kj
s 2 2 @z @z 1 @x @y
20 Then assuming that z f x; y has continuous (or sectionally continuous) derivatives in r, (19) can be written in rectangular form as s 2 2 @z @z dx dy 21 x; y; z 1 @x @y In case the equation for S is given as F x; y; z 0, (21) can also be written q Fx 2 Fy 2 Fz 2 dx dy x; y; z jFz j 22 The results (21) or (22) can be used to evaluate (18). In the above we have assumed that S is such that any line parallel to the zaxis intersects S in only one point. In case S is not of this type, we can usually subdivide S into surfaces S1 ; S2 ; . . . ; which are of this type. Then the surface integral over S is de ned as the sum of the surface integrals over S1 ; S2 ; . . . . The results stated hold when S is projected on to a region r on the xy plane. In some cases it is better to project S on to the yz or xz planes. For such cases (18) can be evaluated by appropriately modifying (21) and (22). THE DIVERGENCE THEOREM The divergence theorem establishes equality between triple integral (volume integral) of a function over a region of threedimensional space and the double integral of the function over the surface that bounds that region. This relation is very important in the expression of physical theory. (See Fig. 105.) Divergence (or Gauss) Theorem Let A be a vector eld that is continuously di erentiable on a closedspace region, V, bound by a smooth surface, S. Then r A dV A n dS 23 where n is an outwardly drawn normal. If n is expressed through direction cosines, i.e., n i cos j cos k cos

