Question

Find positive numbers a and b so that the change of variables s=ax,t=by transforms the integral ∫∫rdxdy into ∫∫t∣∣∣∂(x,y)∂(s,t)∣∣∣dsdt for the region r, the rectangle 0≤x≤15, 0≤y≤60 and the region t, the square 0≤s,t≤1. a= b= what is ∣∣∂(x,y)∂(s,t)∣∣ in this case? ∣∣∂(x,y)∂(s,t)∣∣=

Answer 1

S = ax
It is required to find the value of a which make:
0 ≤ s ≤ 1 when 0 ≤ x ≤ 15
∴ s = 0 when x = 0     ⇒⇒⇒ a = 0 (unacceptable)
or s = 1 when x = 15  ⇒⇒⇒ a = s/x = 1/15

Similarly, t = by
It is required to find the value of b which make:
0 ≤ t ≤ 1 when 0 ≤ y ≤ 60
∴ t = 0 when y = 0     ⇒⇒⇒ b = 0 (unacceptable)
or t = 1 when y = 60  ⇒⇒⇒ b = t/y = 1/60

$a = \frac{1}{15} \ \ \ and \ \ \ b= \frac{1}{60}$

∴ x = 15s  and  y = 60t

$J ( \frac{x,y}{s,t}) = \left[\begin{array}{ccc} \frac{dx}{ds} & & \frac{dx}{dt} \\ & & \\ \frac{dy}{ds} & & \frac{dy}{dt} \end{array}\right] = \left[\begin{array}{ccc}15& &0\\ & & \\0& &60\end{array}\right] = 900$

Note: it is not matrix it is determinant