Advanced Calculus fi..

Chapter 2 Differential Calculus of Functions of Several Variables 125be a curve in the .sllrjuce, passing through (.rl.).I.:I ) when t = rl. One has thusTaking differentials. one concludeswhere JF/J.r, ilF/ily, RF/Bz are to bc evaluated at (.rl, y,. :I) and d.1- = ,/"(tl) dt,dy = gl(tl)rlt, rl~ = hl(tl)dt: as earlier. these differentials can be replaced by.r - .rl. y - yl .:- zI, where (.r. y. :) is a point on the tangent to the given curve at(.rI. yl. :I). Accordingly, (2.99) can be written:This is the equation of a plane containing the tangent linc to the chosen curve.However, (2.100) no longer depends on the particular curve chosen: all tangent linesto curves in the surface through (.rl. yl. :I) lie in the one plane (2.100). which istermed the rrznget~rplane to the srr$ac.e at (.rI, yl. :I). [If all three partial derivativesof F areoat the chosen point, Eq. (2.100) fails to determine a plane, and the definitionbreaks down.]It should be remarked that (2.99) is equivalent to (2.100). that is. that again theoperation of rt~kit~g differentials yielrls the tatlgerzt. As a further illustration of this.consider a curve determined by two intersecting surfaces:Thc corrcsponding differential relationsrepresent two intersecting tangent planes at the poinl (-1-1. yl. :, ) at the point considered;thc intersection of these planes is thc trzngetlt lirzrj to the curve (2.101). Toobtain thc equations in the usual form. the partial derivatives must be evaluated a1(.rl. XI, ). and the differentials d.r, rly, d: must be replaced by 1- --.\-I. y - yl. :-:I.These results can also be put in vector form. First of all. (2.100) shows that thevector (i)F/i).r)i + (i)F/i)v)j + (2FliI:)k is nor711nl to thc tangent plunc at (.ul, j l , :I).This vector is known as the gratlietlt \lcjc.tor of the function F(.r. y. :) (see Fig. 2.13).We writei . i . i)Fgrad F = -I + -J + -k.i1.r ;I)' 0:The notation VF (read "del F") for grad F will also be uscd; this is discussed inthe following section. There is a gradient vector of F at each point of the domainof definition at which the partial derivatives exist: in particular, there is a gradientvector at each point of the surface F(.r, y, ,-) = 0 considered. Thus the gradientvector has a "point of application" and should be thought of as a holltici vector.