# Multivariable Calculus (1st order approximation)

1. Mar 20, 2012

### plzen90

1. The problem statement, all variables and given/known data

Define f:R2→R3 by
f(x,y,z)=(xy+z)
............(x2-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

2. Relevant equations

3. 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)
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)
2-εθ)

(not confident on this)

d)not attempted yet/ dont know how to
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution