Question

If x=rcos⁡θ, y=rsin⁡θ then the value of \(\frac{∂(x,y)}{∂(r,θ)}\) is ________

(a) 1

(b) 0

(c) r

(d) \(\frac{1}{r}\)

This question was addressed to me in quiz.

My question is taken from Jacobians topic in section Partial Differentiation of Engineering Mathematics

Answer 1

The correct answer is (c) r

Explanation: Wkt, \(\frac{∂(x,y)}{∂(r,θ)} = \begin{vmatrix}

\frac{∂x}{∂r} & \frac{∂x}{∂θ}\\

\frac{∂y}{∂r} & \frac{∂y}{∂θ}\\

\end{vmatrix}  \,but\, x=rcos⁡θ, y=rsin⁡θ\)

Thus \( \begin{vmatrix}

cos⁡θ & -rsin⁡θ\\

sin⁡θ & rcos⁡θ\\

\end{vmatrix} = rcos^2 θ + rsin^2 θ = r.\)