Multivariable Calculus Question #2

[psycho81]

Define f : R² -> R³ as

f (x, y) =

( xy )
( y+x²)
( 1 )

Let p = (0, 1)^T and h = (δ,ε)^T

(i) Evaluate f (p) and f (p + h)
(ii) Calculate the Jacobian matrix Df and evaluate Df (p)

(iii) Calculate the first order approximation to f (p + h), namely f (p) + Df (p)h. Show
that the error
e(h) = f (p + h) − [ f (p) + Df (p)h]

satisfies

lim | e(h) | =0
h->0 |h|

 

[micromass]

You know that homework problems belong in the homework forums, right? Just saying...

Anyway, what did you already try to solve this problem? (i) shouldn't be too difficult, it's just substituting numbers in a function...

 

[psycho81]

sorry...new here,

I'm just getting started on this stuff and wondered how you would do this that's all.

 

[micromass]

(i) is really easy. How would you start it? You just put (0,1) in the definition of f...

 

[psycho81]

ok now looking at it it looks like iii) is my only problem which I have no idea how to do.

 

[micromass]

Well, you could start be finding out what exactly the limit is that you need to calculate. Just substitute the e(h), and the f(h) and stuff by it's values obtained in question (i) and (ii).

This should get you started...