THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Mat 225 As we ave seen, te definition of derivative for a Mat 111 function g : R R and for acurveγ : R E n are te same, except for interpretation: (1) g g(a + ) g(a) (a) and γ γ(t 0 + ) γ(t 0 ) (t 0 ), provided te its exist. Wen tey do exist, we say g and γ are differentiable at a R and t 0 R, respectively. By letting x = a + in te first formula and t = t 0 + in te second, tese can be written as (2) g (a) x a g(x) g(a) x a and γ (t 0 ) t t0 γ(t) γ(t 0 ) t t 0. Te difference between tese is tat te numerator is a scalar in te definition of g (a), and is a vector in te definition of γ (a), leading to te interpretation of te slope of te tangent line in te first case and te velocity vector in te second. Now suppose f : E n R or F : E n E k,andletp E n. We want to define wat it means for z = f(x) andz = F (X) to be differentiable at P. A straigt forward mimic of (1) and (2) would be f (P ) =? f(x) f(p ) X P X P f(p + ) f(p ), were = X P is te displacement vector from P to X. Tis is nonsense because we can t divide by vectors. Note tat we can define partial derivatives and directional derivatives in tis way because tey are essentially functions of one variable. Te partial derivative of z = f(x, y) at (x, y) =(a, b) wit respect to x is f x (a, b) x a f(x, b) f(a, b) x a Δx 0 and te derivative of F at P along te vector v is F (P + tv) F (P ) D v F (P ). t 0 t 1 f(a +Δx, b) f(a, b), Δx
Te solution is to realize tat te differentiability of g : R R and γ : R E n can be viewed in anoter way tat can be generalized to f : E n R and F : E n E k. Informally (2) says tat g g(x) g(a) (a) and γ (t 0 ) γ(t) γ(t 0) x a t t 0 wen x is close to a and t is close to t 0.Solvingforg(x) andγ(t) weave (3) g(x) g(a)+g (a)(x a) and γ(t) γ(t 0 )+(t t 0 )γ (t 0 ). In te rigt and sides we recognize y = g(a)+g (a)(x a) as te equation of te tangent line to te grap of g from Mat 111 and X = γ(t 0 )+γ (t 0 )(t t 0 ) as a parameterization of te tangent line to te image of te curve γ. Te idea is tat a function is differentiable if it can be approximated by a linear function. Te functions f : E n R and F : E n E k sould be differentiable at P if tere are good approximations f(x) f(p )+f (P )(X P ) and F (X) F (P )+F (P )(X P ). But wat do tese mean, and ow can we interpret f (P )andf (P )ifwedon tknow wat differentiability sould mean? It s not quite a logical circle. Once we know ow good te approximation needs to be, we can actually replace te slope g (a) andtevelocity vector γ (t 0 ) in (3) wit an arbitrary slope m and vector v, namely if te approximations g(x) g(a)+m(x a) and γ(t) γ(t 0 )+(t t 0 )v are good enoug in a suitable sense, ten m and v are uniquely determined. Wy te Tangent Line is te Best Linear Approximation in te One-Variable Case. An arbitrary non-vertical line troug (a, g(a)) as te form y = g(a)+m(x a), wic you can get by te point-slope formula. Suppose we approximate y = g(x) bytis line near x = a: g(x) g(a)+m(x a). To analyze tis approximation, let s look at te error it makes, namely ɛ = g(x) g(a) m(x a). We want to tink of tis as a function of ow close x is to a, and so we switc to te = x a notation: (4) =g(a + ) g(a) m. 2
If g is continuous at a, ten ( ) g(a + ) g(a) m = g(a) g(a) m0 =0. Te fact tat te error goes to zero as x goes to a is not a big deal... it basically just says tat te curve y = g(x) and te line y = g(a) +m(x a) bot go continuously troug te same point wen x = a. Every line tat goes troug te point (a, g(a)) does tis noting is special about te tangent line. Instead of looking at, wic is zero no matter wat te slope of te line is, let s look at. Since te denominator goes to zero, tis will be less likely to go to 0. Assuming te it in (1) exists, we ave g(a + ) g(a) m g(a + ) g(a) m = g (a) m. Notice tat tis is zero if and only if m = g (a). In oter words, =0ifandonlyif te line is wat was defined in Mat 111 to be te tangent line. Here is te interpretation of = 0 tat I like. Not only is te error going to zero, it is going to zero muc faster tat is going to zero, tat is, muc faster tan x is approacing a. We ave derived tis under te assumption tat g is differentiable at x = a, but tis is really unnecessary. Let s not assume tat g is differentiable, but only tat tere is some line for wic g(a + ) g(a) =0. Ten ( ) g(a + ) g(a) m + m + m = m, and so g turns out to be differentiable after all, and te value of g (a) is te slope of te line. Wat we ave found is summarized in te following teorem. Teorem. Let g : R R. Consider te approximation g(x) g(a)+m(x a) near x = a and its error =g(a + ) g(a) m. Teng is differentiable at a if and only if tere is some slope m suc tat te error goes to zero so quickly tat =0. Furtermore, if g is differentiable at a, teslopem is unique, and is equal to g (a) as defined in (1). A similar teorem olds for curves. 3
Teorem. Let γ : R E n. Consider te approximation γ(t) γ(t 0 )+(t t 0 )v near t = t 0 and its error =γ(t + t 0 ) γ(t 0 ) v. Ten γ is differentiable at t 0 if and only if tere is some vector v suc tat te error goes to zero so quickly tat =0. Furtermore, if γ is differentiable at t 0, te vector v is unique, and is equal to γ (t 0 ) as defined in (1). Note tat te error is a vector in tis case, and it measures ow well te parameterized line X = γ(t 0 )+(t t 0 )v approximates γ(t). Tis parameterized line goes troug te rigt point γ(t 0 )atterigttimet 0, but could be going in te wrong direction or at te wrong speed. Only one line could possibly ave te same speed and direction as γ, te one wit velocity γ (t 0 ). A key point in tese teorems is tat tey caracterize te differentiability of g and γ witout aving to mention g (a) andγ (t 0 ) (te last clause could be left off te statements witout canging te teorems). One additional observation, wic will elp make tis work in more variables: = 0 if and only if = 0 if and only if =0. Te Best Linear Approximation in More Variables. Wat can replace te approximations g(x) g(a)+m(x a) and γ(t) γ(t 0 )+(t t 0 )v in te cases of f : E n R and F : E n E k? For f : E n R, te appropriate approximation is f(x) f(p )+w (X P ), in wic w is a vector on E n and w (X P ) is dot product. As muc as I like doing tings witout coordinates, te coordinate version of F : E n E k is easier to state. For F : R n R k (note te cange from E n to R n ), te appropriate approximation is F (X) F (P )+M(X P ), in wic M is a k n matrix and M(X P ) is matrix multiplication. Part of te statements in te teorems for te one-variable case become definitions in te several-variable case. Definition. Let f : E n R. Consider te approximation f(x) f(p )+w (X P ) near X = P and its error =f(p + ) f(p ) w. Wesaytatf is differentiable at P if tere is some vector w suc tat te error goes to zero so quickly tat 4 =0.
Definition. Let F : R n R k. Consider te approximation F (X) F (P )+M(X P ) near X = P and its error =F (P + ) F (P ) M, werem is a k n matrix. We say tat F is differentiable at P if tere is some matrix M suc tat te error goes to zero so quickly tat =0. (Note tat te error is a vector in tis case.) Te furtermore parts of te teorems are still teorems. Teorem. Suppose f : E n R is differentiable at P. Ten te vector w in te definition of differentiability is unique. In a coordinate system, te coefficients of w are te partial derivatives of f at P. Teorem. Suppose F : R n R k is differentiable at P. Ten te matrix M in te definition of differentiability is unique. Te columns of M are te partial derivatives of F at P. (Remember tat te partial derivatives of F are vectors.) Anyting tat is unique deserves a name and some notation. Definition. If f : E n R is differentiable at P, ten te vector w in te definition of differentiability is called te gradient of f at P, and is denoted by f(p ). Definition. If F : R n R k is differentiable at P, ten te matrix M in te definition of differentiability is called te derivative matrix of F at P, and is denoted by F (P ). You sould be getting te idea tat differential calculus is te matematics of finding linear approximations to functions. If a function is differentiable at a point, it as a unique best linear approximation. Here is a summary: g : R R, g(x) g(a)+g (a)(x a) (5) γ : R E n, γ(t) γ(t 0 )+(t t 0 )γ (t 0 ) f : E n R, f(x) f(p )+ f(p ) (X P ) F : R n R k, F(X) F (P )+F (P )(X P ) I ope it is clear tat you want to view tese as being variations on a common teme, rater tan different. 5
Differentiability and Directional Derivatives. One important fact about differentiability is tat it unifies directional derivatives, or rater, derivatives along vectors. Teorem. Suppose f : E n R is differentiable at P.Ten D v f(p )= f(p ) v. Teorem. Suppose F : R n R k is differentiable at P.Ten D v F (P )=F (P )v. Differentiability and Tangent Lines and Planes. Te linear approximations in (5) lead directly to equations of tangent lines and planes of te objects defined by te functions and mappings. Te two familiar examples are graps of g : R R and parameterized curves γ : R E n. Recall tat for a parameterized curve, te associated object is its image. Function Object Tangent Object g : R R γ : R E n Curve Grap of y = g(x) Parametric curve Image of X = γ(t) Line Grap of y = g(a)+g (a)(x a) Parametric line Image of X = γ(t 0 )+(t t 0 )γ (t 0 ) Te main idea is tat te tangent object bears te same relationsip to its equation as te object bears to its equation, tat is, te tangent object to a grap is te grap of te linear approximation, te tangent object to a parameterized image is te parameterized image of te linear approximation, and te tangent object to a level set is an appropriate level set of te linear approximation. 6
Function Object Tangent Object f : R 2 R f : R 2 R f : R 3 R F : R 2 E 3 Surface Grap of z = f(x) Curve Level set of z = f(x) for const. z = f(p ) Surface Level set of w = f(x) for const. w = f(p ) Parametric surface Image of X = F (T ) were T =(s, t) Plane Grap of z = f(p )+ f(p ) (X P ) Line Level set of z = f(p )+ f(p ) (X P ) for const. z = f(p ) Simplifies to 0 = f(p ) (X P ) Plane Level set of w = f(p )+ f(p ) (X P ) for const. w = f(p ) Simplifies to 0 = f(p ) (X P ) Parametric plane Image of X = F (T 0 )+F (T 0 )(T T 0 ) were T 0 =(s 0,t 0 ) Note tat te tangent object equations for level sets simplify. For f : R 2 R, you get a level curve by setting z equal to a constant in z = f(x). If you want te level curve troug P, te constant is f(p ). To get te tangent line, you set z equal to te same constant in z = f(p )+ f(p ) (X P ). Ten f(p ) cancels from bot sides of te equation, leaving 0 = f(p ) (X P ). Te last item in te cart needs furter discussion to make te tangent plane look parametric. Te parametric plane sould be of te form X = F (s 0,t 0 )+(s s 0 )v+(t t 0 )w, were v and w avesometingto do wit te partialderivatives off. Indeed, te columns of te matrix F (T 0 )=F (s 0,t 0 ) are te partial derivatives, and so te matrix product becomes ( ) F (T 0 )(T T 0 )=F s s0 (s 0,t 0 ) = ( F t t s (s 0,t 0 ) F t (s 0,t 0 ) ) ( ) s s 0 0 t t 0 =(s s 0 )F s (s 0,t 0 )+(t t 0 )F t (s 0,t 0 ). Te parametric plane is ten X = F (s 0,t 0 )+(s s 0 )F s (s 0,t 0 )+(t t 0 )F t (s 0,t 0 ). Examples will be fortcoming in N:/Mat/Mat225/CurvesAndSurfaces.nb. Robert L. Foote, Fall 2007 7