Chapter 14.7, Problem 43ES

(a) Verify that

$\left|\begin{array}{cc}{a}_1& b_1\\ c_1& d_1\end{array}\right|\left|\begin{array}{cc}{a}_2& b_2\\ c_2& d_2\end{array}\right|=\left|\begin{array}{c}{a}_1{a}_2+b_1{c}_2{a}_1{b}_2+b_1{d}_2\\ c_1{a}_2+d_1{c}_2{c}_1{b}_2+d_1{d}_2\end{array}\right|$

(b) If $x=x\left(u,\upsilon \right),y=y\left(u,\upsilon \right)$ is a one-to-one transformation, then $u=u\left(x,y\right),\upsilon =\upsilon \left(x,y\right).$ Assuming the necessary differentiability, use the result in part (a)and the chain rule to show that $\frac{\partial \left(x,y\right)}{\partial \left(u,\upsilon \right)}\cdot \frac{\partial \left(u,\upsilon \right)}{\partial \left(x,y\right)}=1$