UNDAMENTAL CONCEPTS: SURACES AND CURVES CHAPTER udametal Cocepts: Surfaces ad Curves. INTRODUCTION This chapter describes two geometrical objects, vi., surfaces ad curves because the pla a ver importat role i the stud of partial differetial equatios.. SURACES A poit P(,, ) i three-dimesioal space is said to lie o a surface if the coordiates,, are coected b a relatio of the form (,, ) 0...() where is a cotiuousl differetiable fuctio defied o a domai i R 3. Cosider ow a set of relatios of the form: (u,v), (u,v), 3 (u,v)...() The for each pair of values of u ad v there correspod three umbers,, ad so a poit (,, ) i space. However, ever poit i sapce does ot correspod to a pair u ad v. If the Jacobia: (, ) u v 0, ( uv, ) u v the the first two equatios of () ca be solved to epress u ad v as fuctios of ad locall, sa u f (, ) ad v f (,) b the iverse fuctio theorem. Thus u ad v are determied oce ad are kow ad so the third equatio i () gives a value of for these values of ad, i.e., 3 (f (, ), f (, )). Hece there eists a fuctioal relatio betwee the coordiates, ad as i (). Therefore, a poit (,, ) determied b () lies o a fied surface. Equatios of the tpe () are called parametric
A TEXTBOOK O PARTIAL DIERENTIAL EQUATIONS equatios of the surface. However, parametric equatios of a surface are ot uique as eplaied i the followig illustratio. Illustratio : The parametric equatios: av av a si u cos v, a si u si v, a cos u, ad si u, cos u, a ( v ), + v + v + v ad a ( v ) si u, a ( v ) av + v cos u,, where a is a costat, all represet the same + v + v surface + + a which is a sphere..3 CURVES IN SPACE A curve i three-dimesioal space ma be specified b meas of parametric equatios: g (t), g (t), g 3 (t)...(3) where g, g ad g 3 are cotiuous fuctios of a cotiuous variable t which varies i a iterval I R. O elimiatig t betwee g ad g, we obtai a relatio of the form: φ (, ) 0. Similarl, from g st ad g 3 we get (, ) 0. Therefore a curve ca be thought of as the poits of itersectio of two surfaces. Let us cosider a poit P o the curve: (s), (s), (s)...(4) characteried b the arc legth (i.e. legth of the curve) s measured from some fied poit P 0 alog the curve. Let Q (s + ) ( (s + ), (s + ), (s + )) be aother poit o the curve at a distace (alog the curve) from P. Let the chord PQ (see the adjacet figure). The lim P Q. Now the directio cosies of the chord PQ are s ( + ) () s s ( + ) ( s) s ( + ) () s as Q P (i.e., 0). {( s+ ) ( s )} d ; { s ( + s) ( s)} d ; {( s + s ) ( s )} d Q(s+ s) Also the chord PQ takes up the directio to the taget to the curve at P ad hece the directio d d d cosies of the taget to the curve (4) at P are,,. It is oted that the parameter t of (3) is the arc legth if g + g + g 3, where prime deotes differetiatio with respect to t. c ig. P 0 s P(s).4 SURACES AND CURVES Suppose that the curve : ( (s), (s), (s) lies o the surface S : (,, ) 0. The
UNDAMENTAL CONCEPTS: SURACES AND CURVES 3 ((s), (s), (s)) 0, "s...(5) Differetiatig both sides of (5) with respect to s, we get d + d + d 0... () This is the coditio that the taget to the curve Γ at the poit P (,, ) is perpedicular to the lie havig directio ratios,,. The curve Γ is arbitrar ecept for the fact that it passes through the poit P ad lies o the give surface S. Therefore, we coclude that the lie havig directio ratios,, is perpedicular to the taget to ever curve lig o S ad passig through P. Hece (,, ) give the directio ratios of the ormal to the surface S at P. Observatio : A surface ma be formed as beig geerated b a curve. A poit P (,, ) whose coordiates satisf (,, ) 0 ad which lies o the plae k has its coordiates satisfig the equatios: k ; (,, k) 0...(7) epresses the fact that the poit P (,, ) lies o a curve, sa Γ k o the plae k (see ig. ). ig. Illustratio : Cosider S : + + a which is a sphere, the poits of S with k have Γ k : k, + a k Here Γ k is a circle of radius a k which is real if k< a. As k varies from a to + a, each poit of the sphere S is covered b oe such circle Γ k ad so we ma thik of the surface of the sphere as beig geerated b such circles. Note: I geeral we ca thik of the surface (,, ) 0 as beig geerated b the curves Γ k : k, (,, k) 0. Observatio : Two surfaces S : (,, ) 0 ad S : (,, ) 0, will, i geeral itersect i a curve Γ give b Γ : (,, ) 0, (,, ) 0...(8) which is the locus of a poit whose coordiates satisf (,, ) 0 ad (,, ) 0 simultaeousl. If the equatio of the surface S is of the form f (, ) ad if we deote
4 A TEXTBOOK O PARTIAL DIERENTIAL EQUATIONS the sice (,, ) f (, ), it follows that p, q...(9) p, q ad. Therefore directio cosies of ormal to the surface S at the poit (,, ) are p q,,, where p q + +...(0) The equatio of the taget plae π at the poit P (,, ) to the surface S : (,, ) 0 is π : (X ) + + ( Y ) ( Z ) 0...() where (,, ) are the coordiates of a other poit of the taget plae p. Similarl, the equatio of the taget plae p at P (,, ) to the surface S is π : (X ) ( Y ) ( Z ) 0...() S L P S ig. 3 The itersectio L of the plaes p ad p is the taget at P (,, ) to the curve Γ give b (8), see ig. 3. It follows from equatios () ad () that the equatios of the lie L are X ad so the directio ratios of the lie L are Y Z...(3) (, ), (, ) (, ), (, ) (, )....(4)
UNDAMENTAL CONCEPTS: SURACES AND CURVES 5 ILLUSTRATIVE EXAMPLES Eample : Determie the taget vector at 0,, π to the heli give b the followig parametric equatios: cos t, si t, t, t R. d d d Solutio: The taget vector to the heli at t is dt, dt, dt ( si t, cos t, ). Observe that the poit 0,, π correspo to t π give heli is (, 0, ). Eample : id the equatios of the taget lie to the space circle at the poit,, + +, + + 0. Solutio: The give space circle is described as (,, ) + + 0 (,, ) + + 0 ( ) Observe that the give poit /, /, / the taget lie at this poit ca be writte as (see eq. (3): / (, ) + / / (, ) (, ) ( ) where the Jacobias are calculated at /, /, / (, ) (,) (, ) ad so at this poit the taget vector to the lies o this space circle ad so the equatios of ( ), ( ) 0, Therefore the equatios of the required taget lie are / + / / 0 ( ). or, / + / /. 0
A TEXTBOOK O PARTIAL DIERENTIAL EQUATIONS Eample 3: Show that the coditio that the surfaces (,, ) 0, (,, ) 0 should touch is that the elimiat of,, from these equatios ad the equatios should hold. Hece determie the coditio that the plae l + m + p should touch the cetral coicoid a + b + c. Solutio: irst Part: If the give two surfaces touch each other at some poit, sa P (,, ), the these two surfaces have the commo taget plae, i.e., the equatios: ( X ) ( Y ) ( Z ) 0 ad ( X ) ( Y ) ( Z ) 0 must be same ad so we have...() Therefore, the required coditio is obtaied b elimiatig,, from the give equatios of surfaces ad the equatios (). Secod Part: Let (,, ) l + m + p 0...() ad (,, ) a + b + c 0...(3) If these two surfaces touch, the b (): l a m l m k( sa) b c a b c l ak m,, bk ck...(4) rom () ad (4), we have l ak + m bk l m + p k + + ck p a b Agai from (3) ad (4), we have l ak c m l m + + k + + bk ck a b c...(5) p l m + + a b c l m + + [Usig (5)] a b c
UNDAMENTAL CONCEPTS: SURACES AND CURVES 7 p l m + +, this is the required coditio. a b c EXERCISE-. Show that the directio cosies of the taget to the coic a + b + c, + + are proportioal to (b c, c a, a b).. Determie the equatio of the taget lie to the space circle: + +, + + 0 at the poit ( / 4, / 4, 3/ 4). [As. / 4 0 / 4 / 4 + 3/ 4.] 8/ 4 / 4