Question

Evaluation of \(\int\int_R f(x,y) \,dx \,dy \) in cartesian coordinate can be done using change of variables principle, among the choices given below which is correct explanation of change of variables principle? (Given let x=g(u,v) & y=h(u,v))

(a) \(\int\int_S f(g(u,v),h(u,v)) \,du \,dv\)

(b) \(\int\int_S f(g(u,v),h(u,v)) \frac{d(x,y)}{d(u,v)} \,du \,dv\)

(c) \(\int\int_S f(g(u,v),h(u,v)) \frac{∂(x,y)}{∂(u,v)} \,du \,dv\)

(d) \(\int\int_S f(g(u,v),h(u,v)) \frac{∂(u,v)}{∂(x,y)} \,du \,dv\)

I got this question in examination.

I would like to ask this question from Change of Variables In a Double Integral topic in portion Multiple Integrals of Engineering Mathematics

Answer 1

The correct choice is (c) \(\int\int_S f(g(u,v),h(u,v)) \frac{∂(x,y)}{∂(u,v)} \,du \,dv\)

For explanation I would say: \(\int\int_S f(g(u,v),h(u,v)) \frac{∂(x,y)}{∂(u,v)}  \,du \,dv\)

where \( \frac{∂(x,y)}{∂(u,v)} = \begin{vmatrix}

\frac{∂x}{∂u} \frac{∂x}{∂v}\\

\frac{∂y}{∂u} \frac{∂y}{∂v}\\

\end{vmatrix} = J(\frac{x,y}{u,v}) =  \frac{∂x}{∂u}  \frac{∂y}{∂v} – \frac{∂x}{∂v} \frac{∂y}{∂u} …..\)(‘J’ is Jacobian).