If u=yz÷x,v=zx÷y,w=xy÷z find d(u,v,w)/d(x,y,z)?

Question

Answer

Explanation:

To find d(u,v,w)/d(x,y,z), we need to take the partial derivative of each variable with respect to x, y, and z.

Partial derivatives of u:

du/dx = -yz/x^2

du/dy = z/x

du/dz = y/x

Partial derivatives of v:

dv/dx = z/y

dv/dy = -zx/y^2

dv/dz = x/y

Partial derivatives of w:

dw/dx = y/z

dw/dy = x/z

dw/dz = -xy/z^2

Now, we can find d(u,v,w)/d(x,y,z) by arranging the partial derivatives in a matrix and taking the determinant.

The Jacobian:
J = | du/dx du/dy du/dz |
| dv/dx dv/dy dv/dz |
| dw/dx dw/dy dw/dz |

Taking the determinant of this matrix, we get:

d(u,v,w)/d(x,y,z) = x^(-2)yz^(-2) + y^(-2)zx^(-2) + z^(-2)xy^(-2) - y^(-1)z^(-1) - x^(-1)z^(-1) - x^(-1)y^(-1)

Conclusion:

The final answer for d(u,v,w)/d(x,y,z) is (x^(-2)yz^(-2) + y^(-2)zx^(-2) + z^(-2)xy^(-2) - y^(-1)z^(-1) - x^(-1)z^(-1) - x^(-1)y^(-1)).

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