91影视

The equations of transformations are,

$x=伪u+尾v;y=纬u+未v$

The objective is to determine the transformation of a line $ax+by=c$in xy-plane to uv-plane.

The definition of a Jacobian of transformation using partial derivatives is given as

$$\begin{gathered} \frac{鈭�(x,y)}{鈭�(u,v)}=\left|\begin{array}{cc}\frac{鈭�x}{鈭�u}& \frac{鈭�y}{鈭�u}\\ \frac{鈭�x}{鈭�v}& \frac{鈭�x}{鈭�v}\end{array}\right| \\ \frac{鈭�(x,y)}{鈭�(u,v)}=\left|\begin{array}{cc}伪& 纬\\ 尾& 未\end{array}\right| \\ \frac{鈭�(x,y)}{鈭�(u,v)}=伪未-尾纬 \end{gathered}$$

To form the equation of the line in UV- plane , substitute the equations of transformations in the equation of line.

$$\begin{gathered} ax+by=c \\ (a伪+b纬)u+(a尾+b未)v=c \end{gathered}$$

If the above equation has to represent a point in uv- planes, both the coefficients of variables u and v have to be zero.

Let's assume the converse.

$$\begin{gathered} a伪+b纬=0 \\ a尾+b未=0 \end{gathered}$$

Consider the ratio of two statements.

$$\begin{gathered} \frac{伪}{尾}=\frac{纬}{未} \\ 伪未-尾纬=0 \end{gathered}$$

This is the Jacobian of the transformation as determined above.

Thus, it is proved that if the Jacobian of transformation is equal to zero, then the coefficients of both variables u and v in the transformed equation are also zero.

Hence, for the equation to represent a point both the coefficients should be zero, and hence the Jacobian should be also zero.