**Overview**(Jacobian)

Jacobian is defined as- If u and v are functions of the two independent variables x and y , then the determinant,

is known as the jacobian of u and v with respect to x and y.

We can write it as below

Suppose there are three functions u, v and w of three independent variables x , y and z then,

Suppose there are three functions u , v and w of three independent variables x , y and z then,

The Jacobian can be defined as,

**Important properties of the Jacobians**

**Property-1-**

If u and v are the functions of x and y , then

Proof- Suppose u = u(x,y) and v = v(x,y) , so that u and v are the functions of x and y,

Now,

Interchange the rows and columns of the second determinant, we get

Differentiate u = u(x,y) and v= v(x,y) partially w.r.t. u and v, we get

Putting these values in eq.(1) , we get

Hence proved.

**Property-2:**

Second property is also known as chain rule.

Suppose u and v are the functions of r and s, where r,s are the fuctions of x , y, then,\

Interchange the rows and columns in second determinant

We get,

Similarly we can prove for three variables.

**Property-3**

If u,v,w are the functions of three independent variables x,y,z are not independent , then,

Proof: here u,v,w are independent , then f(u,v,w) = 0 ……………….(1)

Differentiate (1), w.r.t. x, y and z , we get

Eliminate from 2,3,4 , we get

Interchanging rows and columns , we get

So that

**Example: If u = x + y + z ,uv = y + z , uvw = z , then find **

Sol. Here we have,

x = u – uv = u(1-v)

y = uv – uvw = uv( 1- w)

And z = uvw

So that,

We get

= u²v(1-w) + u²vw

= u²v

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