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\bf{Question}

Verify the chain rule

$$D(F\circ G)(p)=DF(q).DG(p),\ q=G(p)$$

where $F,G$ are the maps in question 1(ii), (i) respectively. Do
likewise for the maps in (iii),(ii).




\bf{Answer}

With $F$, $G$ as in (ii), (i) respectively we have $$F \circ
G(x_1,x_2) = (x_2^2+2x_2+ 4\sin x_1x_2 + 3(x_1-x_2)^2,
x_2^2+8x_2+10 \sin x_1 x_2 + 6(x_1 - x_2)^2)$$ so $$D(F \circ
G)(x_1, x_2) = $$ $$\left ( \begin{array}{cc} 4x_2 \cos x_1 x_2 +
6(x_1- x_2) & 2x_2 + 2 + 4x_1 \cos x_1 x_2 - 6(x_1-x_2) \\ 10x_2
\cos x_1 x_2 + 12(x_1 - x_2) & 8x_2 + 8 + 10x_1 \cos x_1 x_2 -
12(x_1 - x_2) \end{array} \right )$$

which is the same as the product of matrices in (ii), (i).

Likewise for $F$, $G$ as in (iii), (ii) respectively we have
\begin{eqnarray*}
F \circ G(x_1,x_2,x_3) & = & 2(x_1+2x_2+3x_3)^2 + (x_1+ 2x_2 +
3x_3)( 4x_1 + 5x_2 + 6x_3)\\ & & - (4x_1 + 5x_2 + 6x_3)^2\\ & = &
-10x_1^2 - 7x_2^2 - 19x_1x_2 -9x_2x_3 -18x_1x_3
\end{eqnarray*}

So
\begin{eqnarray*}
D(F\circ G)(x_1,x_2,x_3) & = & d(F \circ G)(x_1, x_2, x_3)\\ & = &
(-20x_1-19x_2-18x_3, -14x_2-19x_1-9x_3, -9x_2-18x_1) \\ & = &
DF(y_1, y_2).DG(x_1, x_2, x_3)
\end{eqnarray*}
where $(y_1, y_2) = G(x_1, x_2, x_3)$.

as
\begin{eqnarray*}
DF(y_1, y_2) & = & (4y_1 + y_2, y_1 - 2y_2)\\ & = &
(4(x_1+2x_2+3x_3) + (4x_1+5x_2+6x_3),\\ & &  (x_1+2x_2+3x_3) -
2(4x_1+5x_2+6x_3))\\ & = & (8x_1+13x_2+18x_3, -7x_1-8x_2-9x_3)
\end{eqnarray*}
and $DG$ as in (ii).



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