Theory of ODEs

Question 1

existence and uniqueness

Answer 1

interval and one and only one continuously differentiable function for which IVP holds

Question 2

maximal interval of existence

Answer 2

there is an open maximal interval of solution

Question 3

global flow

Answer 3

all solutions x_a are defined for all time

Question 4

smooth transformation/diffeomorphism

Answer 4

bijective smooth function from E to E with smooth inverse

Question 5

smooth 1-parameter group of transformations

Answer 5

smooth map st
for all t, X(.,t) is a transformation with id=X(.,0)
X(.,t) forms a subgroup with composition by addition t+s

Question 6

divergence

Answer 6

partial dg_1/dx_1 + … + partial dg_n/dx_n

Question 7

state at t

Answer 7

(x(t),x’(t))

Question 8

equation of motion/Newton’s equation

Answer 8

x’‘=f(t,x,x’)

Question 9

first order system

Answer 9

x’=f(t,x)

Question 10

order of equation/system

Answer 10

highest derivative involved

Question 11

autonomous

Answer 11

t does not intervene

Question 12

flow of an autonomous ODE

Answer 12

denote x_a the maximal solution to the ODE with x(0)=a
Γ^t(a)= x_a(t)

Question 13

Jacobean

Answer 13

det(partial derivative of flow)

Question 14

divergence free

Answer 14

divergence=0

Question 15

(i,j)th minor

Answer 15

matrix obtained by deleting ith row and jth column

Question 16

(i,j)th cofactor

Answer 16

determinant of the minor

Question 17

Jacobian matrix

Answer 17

matrix of partial derivatives