ssrs barcode font pdf These techniques are further illustrated below for plane curves and for three space in the problems. in Visual Studio .NET

Generator QR in Visual Studio .NET These techniques are further illustrated below for plane curves and for three space in the problems.

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EVALUATION OF LINE INTEGRALS FOR PLANE CURVES If the equation of a curve C in the plane z 0 is given as y f x , the line integral (2) is evaluated by placing y f x ; dy f 0 x dx in the integrand to obtain the de nite integral a2 Pfx; f x g dx Qfx; f x g f 0 x dx 7
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which is then evaluated in the usual manner. Similarly, if C is given as x g y , then dx g 0 y dy and the line integral becomes b2 Pfg y ; ygg 0 y dy Qfg y ; yg dy
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If C is given in parametric form x  t ; y t , the line integral becomes t2 Pf t ; t g 0 t dt Qf t ; t g; 0 t dt
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where t1 and t2 denote the values of t corresponding to points A and B, respectively. Combinations of the above methods may be used in the evaluation. If the integrand A dr is a perfect di erential, d , then c;d A dr d c; d a; b 6
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Similar methods are used for evaluating line integrals along space curves.
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PROPERTIES OF LINE INTEGRALS EXPRESSED FOR PLANE CURVES Line integrals have properties which are analogous to those of ordinary integrals. P x; y dx Q x; y dy P x; y dx Q x; y dy 1:
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a2 ;b2 2:
a1 ;b1
P dx Q dy
a1 ;b1
a2 ;b2
P dx q dy
Thus, reversal of the path of integration changes the sign of the line integral. a2 ;b2 3:
a2 ;b1
P dx Q dy
a3 ;b3
a1 ;b1
P dx Q dy
a2 ;b2
a3 ;b3
P dx Q dy
where a3 ; b3 is another point on C. Similar properties hold for line integrals in space.
LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS
[CHAP. 10
SIMPLE CLOSED CURVES, SIMPLY AND MULTIPLY CONNECTED REGIONS A simple closed curve is a closed curve which does not intersect itself anywhere. Mathematically, a curve in the xy plane is de ned by the parametric equations x  t ; y t where  and are singlevalued and continuous in an interval t1 @ t @ t2 . If  t1  t2 and t1 t2 , the curve is said to be closed. If  u  v and u v only when u v (except in the special case where u t1 and v t2 ), the curve is closed and does not intersect itself and so is a simple closed curve. We shall also assume, unless otherwise stated, that  and are piecewise di erentiable in t1 @ t @ t2 . If a plane region has the property that any closed curve in it can be continuously shrunk to a point without leaving the region, then the region is called simply connected; otherwise, it is called multiply connected (see Fig. 10-2 and Page 118 of 6). As the parameter t varies from t1 to t2 , the plane curve is described in a certain sense or direction. Fig. 10-2 For curves in the xy plane, we arbitrarily describe this direction as positive or negative according as a person traversing the curve in this direction with his head pointing in the positive z direction has the region enclosed by the curve always toward his left or right, respectively. If we look down upon a simple closed curve in the xy plane, this amounts to saying that traversal of the curve in the counterclockwise direction is taken as positive while traversal in the clockwise direction is taken as negative.
GREEN S THEOREM IN THE PLANE This theorem is needed to prove Stokes theorem (Page 237). Then it becomes a special case of that theorem. Let P, Q, @P=@y; @Q=@x be single-valued and continuous in a simply connected region r bounded by a simple closed curve C. Then   @Q @P P dx Q dy dx dy 10 @x @y C r is used to emphasize that C is closed and that it is described in the positive direction. where
This theorem is also true for regions bounded by two or more closed curves (i.e., multiply connected regions). See Problem 10.10.
CONDITIONS FOR A LINE INTEGRAL TO BE INDEPENDENT OF THE PATH The line integral of a vector eld A is independent of path if its value is the same regardless of the (allowable) path from initial to terminal point. (Thus, the integral is evaluated from knowledge of the coordinates of these two points.) For example, the integral of the vector eld A yi xj is independent of path since x2 y 2 A dr y dx x dy d xy x2 y2 x1 y1
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