Question

if u=ax+by and v=cx+dy then d(u,v)/d(x,y)?

Answer 1

Answer:

For u = ax + by

partial diff w rt x = a

partial diff w rt y = b

for v = cx + dy

partial diff w rt x = c

partial diff w rt y = d

Answer 2

Given,

u=ax+by                                               ............... (1)        

v=cx+dy                                               ................ (2)

To Find,

$\frac{d(u,v)}{d(x,y)}$

Solution,

By definition of Jacobian we have,

$\frac{d(u,v)}{d(x,y)}$ =  $\left[\begin{array}{cc}\frac{du}{dx} &\frac{du}{dy} \\\frac{dv}{dx} &\frac{dv}{dy} \end{array}\right]$                             ................... (3)

So,

$\frac{du}{dx}$ = a                                         [ partially differentiatind equation (1) w.r.t. x]

similarly,

$\frac{du}{dy}$ = b

$\frac{dv}{dx}$ = c

$\frac{dv}{dy}$ = d

From (3),

$\frac{d(u,v)}{d(x,y)}$ = $\left[\begin{array}{cc}a&b\\c&d\\\end{array}\right]$ = ad-bc.