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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):
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CHAP. 10]
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LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS
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  0  @r rF @r @r @r    @v jrFj @x @y @v2  1
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[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
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The formulas (14) and (16) also can be introduced in the following nonvectorial manner. Let S be a two-sided surface having projection r on the xy plane as in the adjoining Fig. 10-4. Assume that an equation for S is z f x; y , where f is single-valued 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 .
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Fig. 10-4
Let  x; y; z be single-valued and continuous at all points of S. Form the sum
n X p 1
 p ; p ; p Sp
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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 z-axis, 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 z-axis 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 three-dimensional 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. 10-5.) Divergence (or Gauss) Theorem Let A be a vector eld that is continuously di erentiable on a closed-space 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
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