Question

If each of u ,v ,w a function of variables x,y,z then the jacobian δ(u,v,w)/ δ(x,y,z) is determinant of order............

Answer 1

If each of u ,v ,w a function of variables x , y , z then the jacobian δ(u,v,w)/ δ(x,y,z) is determinant of order 3

Given :

Each of u , v , w a function of variables x , y , z

To find :

The order of the determinant for the jacobian δ(u,v,w)/ δ(x,y,z)

Solution :

Step 1 of 2 :

Define Jacobian

Here each of u ,v ,w a function of variables x , y , z

Then the Jacobian is defined as below

$\displaystyle \sf{ = \frac{ \partial (u,v,w)}{ \partial (x,y,z)} }$

$= \displaystyle \begin{vmatrix} \frac{ \partial u}{ \partial x} & \frac{ \partial u}{ \partial y} & \frac{ \partial u}{ \partial z}\\ \\ \frac{ \partial v}{ \partial x} & \frac{ \partial v}{ \partial y} & \frac{ \partial v}{ \partial z} \\ \\ \frac{ \partial w}{ \partial x} & \frac{ \partial w}{ \partial y} & \frac{ \partial w}{ \partial z} \end{vmatrix}$

Step 2 of 2 :

Find order of the determinant for the jacobian

We see that for the above Jacobian

Number of rows = 3

Number of columns = 3

Hence order of the determinant for the jacobian δ(u,v,w)/ δ(x,y,z) is 3

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