(read nabla or del) is defined by, k. (9.7.1*)

Transcription

1 9.7 Gadient of a scala field. Diectional deivative Some of the vecto fields in applications can be obtained fom scala fields. This is vey advantageous because scala fields can be handled moe easily. The gadient of a given scala function f ( xyz,, ) denoted by gad f o the vecto function, f (ead nabla f ) is gad f f f f f i j k. (9.7.1) Hee x, yz, ae Catesian coodinates in a domain in -space in which f is defined and diffeentiable. (The symbol is also called del - pg. 8 Hildeband.) Example: If f ( xyz,, ) y4xz xthen gad f (4z) i 6y 4 j xk. The diffeential opeato (ead nabla o del) is defined by, i j k. (9.7.1*) Gadients ae useful in seveal ways, notably in giving the ate of change of f ( xyz,, ) in any diection in space, in obtaining suface nomal vectos, and in deiving vecto fields fom scala fields Diectional deivative Fom calculus we know that,, give the ates of change of f ( xyz,, ) in the diection of the coodinate axes. It seems natual to extend this and ask fo the ate of change of f in an abitay diection in space. The diectional deivative Db f o df / ds of a function f ( xyz,, ) at a point P in the diection of a vecto b is, df f ( Q) f ( P) Db f lim. (9.7.) ds s 0 s Hee Q is a vaiable point on the staight line L in the diection of b, and s is the distance between P and Q. Also s 0 if Q lies in the diection of b (see Fig. 1), s 0 if Q lies in the diection of b. May 0,

2 Next, use Catesian xyz -coodinates and fo b a unit vecto. Then the line L is given by () s x() s i y() s jz() s k p sb, (9.7.) 0 whee p 0 is the position vecto of P. Equation (9.7.) now shows that Db f df / ds is the deivative of the function f ( xs ( ), ys ( ), zs ( )) with espect to the ac length s of L. Hence, assuming that f has continuous patial deivatives and applying the chain ule df f f f D f x y b z, (9.7.4) ds whee pimes denotes deivatives with espect to s (which ae taken at s 0 ). But hee diffeentiating (9.7.) gives () s xi yj zk b. Hence (9.7.4) is simply the inne poduct of gad f and b, df Db f b gad f. (9.7.5) ds (Note that in eqn (9.7.5) b 1.) Example 1: Gadient. Diectional deivative Find the diectional deivative of diection of aik. at P :(,1,) in the f ( xyz,, ) x y z Solution: gad f 4xi 6yjzk, gives at P the vecto gad f( P) 8i 6 j6k. 1 baˆ ( ik) Db f( P) ( i k) (8i6j6 k ). 5 5 The minus sign indicates that at P the function f is deceasing in the diection of a Gadient is a vecto Since gad f is defined in tems of components depending on the Catesian coodinates, to pove that gad f is actually a vecto, we must show that gad f has a length and diection independent f f f of the choice of those coodinates. In contast i j k also looks like a vecto but does not have a length and diection independent of the choice of Catesian coodinates. gad f points in the diection of maximum incease of f. Theoem 1: Vecto chaacte of gadient. Maximum incease Let f ( P) f( x, y, z) be a scala function having continuous fist patial deivatives in some domain B in space. Then gad f exists in B and is a vecto (i.e., its length and diection ae independent of the paticula choice of Catesian coodinates). If gad f( P) 0 at some point P, it has the diection of maximum incease of f at P. May 0,

3 Poof: Fom eqn (9.7.5), Db f b gad f b gad f cos gad f cos, (9.7.6) whee is the angle between b and gad f. Now f is a scala function. Hence its value at a point P depends on P but not on the paticula choice of coodinates. The same holds fo the ac length s of the line L (Fig. 1), hence also fo Db f. Now (9.7.6) shows that Db f is maximum when 0 and then Db f gad f. It follows that the length and diection of gad f ae independent of the choice of coodinates. Since 0 if and only if b has the diection of gad f, the latte is the diection of maximum incease of f at P, povided gad f 0 at P Gadient as suface nomal vecto Let S be a suface epesented by f ( x, y, z) c const, whee f is diffeentiable. Such a suface is called a level suface of f, and fo diffeent c we get diffeent level sufaces. Let C be a cuve on S though a point P of S. As a cuve in space, C has a epesentation () t x() t i y() t jz() t k. Fo C to lie on the suface S, the components of ( t) must satisfy f ( xyz,, ) c, i.e., f ( xt ( ), yt ( ), zt ( )) c. (9.7.7) A tangent vecto of C is () t x() t i y() t jz() t k (pg. 9 EK). The tangent vectos of all cuves on S passing though P will geneally fom a plane, called the tangent plane of S at P. (Exceptions occu at edges o cusps of S, fo instance, fo the cone in Fig. 15 at the apex.) The nomal of this plane (the staight line though P pependicula to the tangent plane) is called the suface nomal to S at P. Diffeentiating eqn (9.7.7) with espect to t, f f f x y zgad f 0. Hence, gad f is othogonal to all the vectos in the tangent plane, so that it is a nomal vecto of S at P. Theoem : Gadient as suface nomal vecto Let f be a diffeentiable scala function in space. Let f ( x, y, z) c const epesent a suface S. Then if the gadient of f at a point P of S is not the zeo vecto, it is a nomal vecto of S at P. May 0,

4 Example : Gadient as suface nomal vecto. Cone Find a unit nomal vecto n of the cone of evolution P :(1,0,). z x y 4( ) at the point Solution: The cone is the level suface f 0 of f ( xyz,, ) 4( x y) z. Thus 1 gad f 8xi 8yjzk, gad f( P) 8i 4k. n ( ik ). 5 The vecto n points downwad since it has a negative z -component. The othe unit nomal vecto of the cone at P is n Vecto fields that ae gadients of scala fields ("Potentials") Some vecto fields have the advantage that they can be obtained fom scala fields, which can be handled moe easily. Such a vecto field is given by a vecto function v( P) which is obtained as the gadient of a scala function, v( P) gad f( P). The function f ( P ) is called a potential function o a potential of v ( P ). Such a v( P) and the coesponding vecto field ae called consevative because in such a vecto field, enegy is conseved; i.e., no enegy is lost (o gained) in displacing a body (o a chage in the case of an electic field) fom a point P to anothe point in the field and back to P. (See Sec. 10..) Consevative fields play a cental ole in physics and engineeing. A basic application concens the gavitational foce (Ex. in Sec. 9.4) and we show that it has a potential which satisfies Laplace's equation, the most impotant patial diffeential equation in physics and its applications. May 0,

5 Theoem : Gavitational field. Laplace's equation The foce of attaction, c c p [( x x ) ( ) ( ) ] 0 y y0 z z i j 0 k, (9.7.8) between two paticles at points P0 :( x 0, y0, z0) and P: ( x, y, z) (as given by Newton's law of gavitation) has the potential f ( xyz,, ) c/, whee ( 0) is the distance between P0 and P. Thus p gad f gad c. This potential f is a solution of Laplace's equation, f f f f 0. (9.7.9) ( f (ead nabla squaed f, o del squaed f ) is called the Laplacian of f.) Poof: The distance is 1/ [( xx ) ( y y ) ( zz ) ] p c c c p i 1 p j p k gad f x i y j z k d 1 cx ( x0 ) d 1 cy ( y0) p1 c, p d x c, d y d 1 cz ( z0) p c d z p given by eqn (9.7.8) is the gadient of the scala function f ( xyz,, ) c/. Conside cx ( x) ( xx) x x x x 5. f f c Similaly f 1 ( y y ) f 1 ( zz ) y z f f f 0. Hence f c, c 5 5. f is also denoted by f. The diffeential opeato, (9.7.11) ead "nabla squaed" o "del squaed" o delta, is called the Laplace opeato. It can be shown that the field of foce poduced by any distibution of masses is given by a vecto function that is the gadient of a scala function f and f satisfies eqn. (9.7.9) in any egion that is fee of matte. May 0,

6 The Laplace equation is vey impotant because thee ae othe laws in physics that ae of the same fom as Newton's law of gavitation. Fo instance in elastostatics the foce of attaction (o epulsion) between two paticles of opposite (o like) chage and Q is, Q1 k p. Coulomb's law (9.7.1) The Laplace equation will be discussed in detail in Ch. 1 and Ch. 18. A method fo finding out whethe a given vecto field has a potential will be explained in Sec May 0,