Given that G(x,y) = (x^2+1, y^2) and F(u,v) = (u + v, v^2). compute the Jacobian derivative matrix of F(G(x,y)) at the point (x,y) = (1,1).
A hint to get started will suffice, thanks!
J(F(G(x,y)))=J(F(u0,v0))J(G(x,y))is the basic theorem of Jacobian Matrix
Let we define a series of functions
f1(x1,x2,...xn)
f2(x1,x2....xn)
...
fm(x1,x2...,xn)
this set of functions will be written as F,where
F(x1,...,xn)=(y1,...ym)
yi=fi(x1,...xn)
then J(F) will be
(f1'x1,f1'x2.....f1'xn
f2'x1,f2'x2.....f2'xn
....
fm'x1,fm'x2,......fm'xn)
it is a n*m matrix
so J(F(G(x,y)))=J(F(u0,v0))J(G(x,y))=
(1,0
0,2v)
*
(2x,0
0,2y)
=(2x,0
0,4vy)
=(2x,0
0,4y^3)
I do not exactly know what you are confusing about....
you may understand J(F(G(x,y)))=J(F(u0,v0))J(G(x,y)) like
(fg)'(x)=f'(g(x))g'(x)
the function G actually put (x,y) to (x^2+1,y^2)
and F put (u,v) to (u+v,v^2)
For the format reason, we may define f(x,y)'x to be the partial deriviation of x, say the lim (f(x+dx,y)-f(x,y))/dx.
if you want to know (FG)(x,y)'x or (FG)(x,y)'y, you may first know F(u,v)'v or F(u,v)'v or G(x,y)'x or G(x,y)'y because it will be easier.
The Jacobian's theorem actually provides you a perfect way to solve the problem.
Let we define a series of functions
f1(x1,x2,...xn)
f2(x1,x2....xn)
...
fm(x1,x2...,xn)
this set of functions will be written as F,where
F(x1,...,xn)=(y1,...ym)
yi=fi(x1,...xn)
then J(F) will be
(f1'x1,f1'x2.....f1'xn
f2'x1,f2'x2.....f2'xn
....
fm'x1,fm'x2,......fm'xn)
it is a n*m matrix
I hope you can understand the definition of Jacobian, although it will be a little abstract..
To get the answer, you may first write (FG)(x,y) as F(g1(x,y),g2(x,y)), where g1(x,y)=x^2+1,g2(x,y)=y^2
In this case, we can let u0=g1(x,y),v0=g2(x,y).
Then we can use our equation J(F(G(x,y)))=J(F(u0,v0))J(G(x,y)).
The first part J(F(u0,v0)) tells you that you can ignore the existence of x,y, just looking at u0 and v0.After calculating, you just let u0=g1(x,y),v0=g2(x,y), then everything is completed.
so J(F(u0,v0))=
((u0+v0)'u0,(u0+v0)'v0
(v0^2)'u0,(v0^2)'v0)
=(1,1
0,2v0)
The second part J(G(x,y)) tells you where the x,y will appear.
so J(G(x,y))=
((x^2+1)'x,(x^2+1)'y
(y^2)'x,(y^2 )'y)
=(2x,0
0,2y)
Then we calculate J(F(u0,v0))J(G(x,y))
it will be
(2x,2y
0,4v0y)
then v0=g2(x,y)=y^2
so the answer will be
(2x,2y
0,4y^3)
=(2,2
0,4)
$\displaystyle Df(u,v) = \begin{bmatrix} 1&1\\0&2v\end{bmatrix} $
$\displaystyle Dg(x,y) = \begin{bmatrix} 2x&0\\0&2y\end{bmatrix} $
when $\displaystyle (x,y)=(1,1) \ , \ (u,v) = (2,1) $
$\displaystyle D f\big(g(1,1)\big) = Df(2,1)Dg(1,1) = \begin{bmatrix} 1&1\\0&2\end{bmatrix}\begin{bmatrix} 2&0\\0&2\end{bmatrix} $ $\displaystyle = \begin{bmatrix} 2&2\\0&4\end{bmatrix} $
That was actually extremely helpful.
I only have this one question to learn what a Jacobian matrix is, so I'm desperate to try some other examples. Do you have some?
Also, what is the Jacobian matrix good for? What's the intuition behind it? Judging by how little material I can find on the web, this is probably a difficult question that not many will be able to answer.
yeah,Jacobian Matrix is very useful especially when the function is very complex. And it will be used in some important theorems multivariable deriviation.(But I do not know its English name..)
If you use the Jacobian, you can get several answer at the same time, and the process will be more easier and clearer.
you may want some exercises??I have some..
1.
f(x,y,z)=(x^2+y+z,2x+y+z^2,0),g(u,v,w)=(uv^2w^2,(w ^2sinv),u^2e^v)
Calculate J(F(G(u,v,w)))
2.
if f(x,y,z)=F(u,v,w), x^2=vw,y^2=wu,z^2=uv
prove:
xf'x+yf'y+zf'z=uF'u+vF'v+wF'w
I hope these two will be helpful....
Okay for the first problem, I get:
$\displaystyle
\left[\begin{array}{ccc}2x&1&1 \\ 2&1&2z \\ 0&0&0 \end{array}\right]*\left[\begin{array}{ccc}v^2w^2&2uvw^2&2uv^2w \\ 0&w^2cosv&2wsinv \\ 2ue^v&u^2e^v&0 \end{array}\right] = \left[\begin{array}{ccc}\end{array}\right]
$
Which should be converted to:
$\displaystyle
\left[\begin{array}{ccc}2uv^2w^2&1&1 \\ 2&1&2u^2e^v \\ 0&0&0 \end{array}\right]*\left[\begin{array}{ccc}v^2w^2&2uvw^2&2uv^2w \\ 0&w^2cosv&2wsinv \\ 2ue^v&u^2e^v&0 \end{array}\right]
$
=
$\displaystyle
\left[\begin{array}{ccc}2uv^4w^4+2ue^v&4u^2v^3w^4+w^2cos v+u^2e^v&4u^2v^4w^3+2wsinv \\ 2v^2w^2+4u^3e^{2v}&4uvw^2+w^2cosv+2u^4e^{2v}&4uv^2 w+2wsinv \\ 0&0&0 \end{array}\right]
$
Is it correct?