- #1

plzen90

- 4

- 0

## Homework Statement

Define f:R

^{2}→R

^{3}by

f(x,y,z)=(xy+z)

...(x

^{2}-yz)

let p = (1,1,1)

^{T}and h=(δ,ε,θ)

a)what are n and m? evaluate f(p) and f(p+h)

b)Calculate the Jacobian Matrix Df(x,y,z) and evaluate Df(p)

c) Calculate the error e(h) in the first order approximation to f(p+h)

d) show clearly that

lim h→0

__|e(h)|__=0

......|h|

Explain why this is what you expect

## Homework Equations

## The Attempt at a Solution

a)

n=3

m=2

f(p)=(1,1,1) = (2)

......(0)

f(p+h) = f(1+δ, 1+ε, 1+θ)

=(2+δε+δ+ε+θ)

(δ

^{2}+2δ-εθ-ε-θ)

b)

jac= (y x 1)

...(2x -z -y)

Df(p)=(1 1 1)

...(2 -1 -1)

c)

f(p+h)≈f(p)+Df(p)h

only calculation of Df(p)h needed to work out error.

=(y+x+1)h

(2x-z-y)

=(ε+δ+1)

(2δ-θ-ε)

e(h)=f(p+h)-(f(p) + Df(p)h)

=(δε+θ-ε-1)

(δ

^{2}-εθ)

(not confident on this)

d)not attempted yet/ don't know how to