Question

Which among the following is the definition of Jacobian of u and v w.r.t x and y?

(a) \(J(\frac{x,y}{u,v})\)

(b) \(J(\frac{u,v}{x,y})\)

(c) \(\frac{∂(x,y)}{∂(u,v)} \)

(d) \(\frac{∂(u,x)}{∂(v,y)} \)

This question was posed to me by my school principal while I was bunking the class.

Enquiry is from Jacobians in section Partial Differentiation of Engineering Mathematics

Answer 1

Right option is (b) \(J(\frac{u,v}{x,y})\)

For explanation I would say: \(J(\frac{u,v}{x,y}) = \frac{∂(u,v)}{∂(x,y)} = \begin{vmatrix}

\frac{∂u}{∂x} & \frac{∂u}{∂y}\\

\frac{∂v}{∂x} & \frac{∂v}{∂y}\\

\end{vmatrix}\)