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 
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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