Smooth functions on R^n

Question 1

differentiable at x

Answer 1

there exists a linear map Dfx: V–>W

Question 2

derivative

Answer 2

f(x+v)=f(x)+Dfx(v)+g(v)||v||

Question 3

differentiable on U

Answer 3

at all x in U

Question 4

C0

Answer 4

f is continuous

Question 5

Ck

Answer 5

f is differentiable and Df is C(k-1)

Question 6

f is C1

Answer 6

all first order partial derivatives exist and are continuous

Question 7

f is smooth

Answer 7

partial derivatives of all orders exist

Question 8

f is C2

Answer 8

D2fx is symmetric

Question 9

Chain rule

Answer 9

U, V open f,g differentiable f(x)=V
//
g o f is differentiable, D(g o f)x= Dg(f(x)) o Dfx

Question 10

Mean value theorem

Answer 10

f differentiable, [x,y] is in U
//
there exists ξ in interval st f(y)-f(x)=Dfξ(y-x)

Question 11

Mean value inequality

Answer 11

||f(x)-f(y)|| <= ||Dfξ(y-x)||=||y-x||sup||Dfξ||op

Question 12

smooth def (for non-open sets)

Answer 12

x has an open neighbourhood U and smooth function F st the restriction of F to UnS equals f

Question 13

diffeomorphism

Answer 13

differentiable and differentiable inverse

Question 14

local diffeomorphism

Answer 14

Dfx is an isomorphism at all x

Question 15

inverse function theorem

Answer 15

f is C1 and Dfx is an isomorphism, x has an open neighbourhood V st W=f(V) is open and the restriction f:V–W is a diffeo

Question 16

contraction mapping theorem

Answer 16

S closed, H:S–S. If there exists c<1 st ||H(y)-H(x)||<=c||y-x|| it has a unique fixed point