Find the Laplace equation value of the following potential field V = x2 – y 2 + z2

To find the Laplace equation value of the potential field ( V = x^2 – y^2 + z^2 ), we need to compute the Laplacian operator ( nabla^2 V ) in three-dimensional Cartesian coordinates.

The Laplacian in Cartesian coordinates is given by:

[

nabla^2 V = frac{partial^2 V}{partial x^2} + frac{partial^2 V}{partial y^2} + frac{partial^2 V}{partial z^2}

]

Now, we compute the second derivatives of ( V ):

1. First Partial Derivative with respect to ( x ):

[

frac{partial V}{partial x} = 2x

]

Second Partial Derivative with respect to ( x ):

[

frac{partial^2 V}{partial x^2} = 2

]

2. First Partial Derivative with respect to ( y ):

[

frac{partial V}{partial y} = -2y

]

Second Partial Derivative with respect to ( y ):

[

frac{partial^2 V}{partial y^2} = -2

]

3. First Partial Derivative with respect to ( z ):

[