Question

The jacobian of p,q,r  w.r.t  x,y,z  given p=x+y+z, q=y+z, r=z is ________

(a) 0

(b) 1

(c) 2

(d) -1

The question was posed to me during an internship interview.

The query is from Jacobians in portion Partial Differentiation of Engineering Mathematics

Answer 1

Correct choice is (b) 1

To explain I would say: We have to find

\(J = \frac{∂(p,q,r)}{∂(x,y,z)} = \begin{vmatrix}

\frac{∂p}{∂x} & \frac{∂p}{∂y} &\frac{∂p}{∂z}\\

\frac{∂q}{∂x} &\frac{∂q}{∂y} &\frac{∂q}{∂z}\\

\frac{∂r}{∂x} &\frac{∂r}{∂y} &\frac{∂r}{∂z}\\

\end{vmatrix}\)

But p=x+y+z, q=y+z, r=z (taking partial derivative)

\(J=\begin{vmatrix}

1&1&1\\

0&1&1\\

0&0&1\\

\end{vmatrix}(\frac{∂p}{∂x}=1, \frac{∂p}{∂y}=1, \frac{∂p}{∂z}=1, \frac{∂q}{∂x}=0, \frac{∂q}{∂y}=1, \frac{∂q}{∂z}=1, \frac{∂r}{∂x}=0, \\

\frac{∂r}{∂y}=0, \frac{∂r}{∂z}=1)\)

On expanding we get

J = 1(1 – 0) = 1

Thus j = 1.