Chapter One: Calculus Revisited

Transcription

1 Chpter One: Clculus Revisited 1 Clculus of Single Vrible Question in your mind: How do you understnd the essentil concepts nd theorems in Clculus? Two bsic concepts in Clculus re differentition nd integrtion Let f(t) be function on vrible t The differentition is denoted by df dt or f (t), nd the (definite) integrtion by f(t) dt ( nd b re numbers) Differentition is usully relted to concepts like tngent, grdient or slope, rte, infinitesiml chnge, etc While integrtion is relted to notions like re, volume, totlity, etc Two useful formuls in differentition re the product formul (11) (f g) = f g + g f nd the chin rule (12) (f g(x)) = f (g(x)) g (x) Differentition nd integrtion re not totlly independent concepts, rther they re inverse/reflection of ech other Their explicit reltion is given by (13) f (t) dt = f(b) f(), b, which is known s the fundmentl theorem of clculus 1

2 Theorem 14 (Integrtion by Prts) Let f nd g be differentible on [, b] nd f nd g be integrble on [, b] We hve fg dt = f(b)g(b) f()g() f g dt Theorem 15 (Substitution Formul) Let f be integrble on [, b] nd g differentible function on [c, d] with g > 0 in [c, d] Denote by = g(c) nd b = g(d) We hve d c f(g(t)) g (t) dt = f(s) ds Theorem 16 (Condition for Mx/Min) Let f be differentible function on (, b) If c (, b) is locl mximum for f, then f (c) = 0 nd f (c) 0 Question: Formulte the corresponding sttement for locl minimum 2

3 2 Clculus of Severl Vribles Question in your mind: How to generlize the results in the bove section to functions of more thn one vrible? Let Ω IR n be non-empty open set nd f : Ω IR function For point ( x 1, x 2,, x n ) Ω, the prtil derivtive f/ x i (1 i n) is defined by f f( x 1, x 2,, x i + t,, x n ) f( x 1, x 2,, x i,, x n ) ( x 1, x 2,, x n ) = lim, x i t 0 t provided the limit exists The grdient of f is given by f = ( f, f,, f ) x 1 x 2 x n Exmple Find the grdient of the function f(x 1,, x n ) = (x 1,, x n ) (0,, 0) 1 x x 2 n for Let u be non-zero vector in IR n The directionl derivtive of f in the direction u is given by Exmple f n D u f = f u u Exercise For the function f(x, y, z) = x cos (yz), find the directionl derivtive of f t (1, 2, 3) in the direction of u = (1, 2, 1) 3

4 Higher derivtives re denoted by n f x n 1 1 x n 2 2 x nn n where n 1,, n n re non-negtive integers nd n = n 1 + n n n In this course we del minly with prtil derivtives of first nd second order only Of prticulr interest is the Lplcin of f: f = 2 f x f x 2 2, f x 2 n Theorem 21 (Conditions for Mx/Min) Let f : Ω IR be smooth function If ( 1, 2,, n ) Ω is locl mximum, then f( 1, 2,, n ) = 0 (necessry condition) Sufficient condition: if f( 1, 2,, n ) = 0 nd the Hessin mtrix ( 2 ) f ( 1, 2,, n ) x i x j 1 i,j, n is negtive definite, then ( 1, 2,, n ) is locl mximum for f Note tht in this cse f, which is the trce of the Hessin mtrix, is negtive Question: Find the corresponding sttement for locl minimum Theorem 22 (Chin Rule/Chnge of Vribles) Let f : Ω IR be smooth function Assume tht x 1,, x n re functions of u 1,, u m We hve s function of u 1,, u m Then f(x 1 (u 1,, u m ), x 2 (u 1,, u m ),, x n (u 1,, u m )) f = f x 1 x 1 + f x 2 x f x n x n Similr expressions for f u i, i = 2, 3,, n (try to write them down) 4

5 Multiple Integrtion Question: How would we do integrtion in higher dimension? Do we hve similr formuls s chnge of vribles nd integrtion by prts? Let γ : [, b] Ω IR n be curve nd f : Ω R is function The line integrl is given by f(γ(t)) γ (t) dt (As the functions re ssumed to be smooth in most cses of the course, we re not concerned with the question on the existence of differentition nd integrtion, unless otherwise specificlly mentioned) Question: Let Wht is the mening of the term γ (t)? F (x 1,, x n ) = (f 1 (x 1,, x n ),, f n (x 1,, x n )) be vector field defined on Ω, where f 1,, f n re functions on Ω Denote γ F dγ = F γ (t) γ (t) γ (t) dt Let Ω be non-empty open domin in IR n 1 Let f : Ω IR be smooth function Consider the (hyper)-surfce S IR n+1 given by S = {(x 1,, x n 1, f(x 1,, x n 1 )) (x 1,, x n ) Ω} A norml N to the surfce S is given by N = ( f, f,, f, 1) x 1 x 2 x n 1 Let g : IR n IR be smooth function The surfce integrl is denoted by S g ds = = g N dx 1 dx 2 dx n 1 Ω ) 2 ( ) 2 ( ) 2 ( f f f g dx 1 dx 2 dx n 1 Ω x 1 x 1 x n Question Guess the mening of the term N 5

6 Let F be vector field defined on IR n We hve N F ds = F S S N ds = F N dx1 dx 2 dx n 1 Ω The divergence of vector field F = (f 1,, f n ) is given by div F = f 1 x 1 + f 2 x 2 + f n x n We now look t generliztion of the integrtion by prts formul Theorem 31 (Green s Theorem) Let Ω be bounded domin in IR n nd F be smooth vector field defined in Ω Let S be the boundry of Ω We hve divf dv = F ds Ω S 6