# Consider the Vector field F(x,y,z) equals (6y, 6x, z). Show that F is a gradient vector field F...

## Question:

Consider the Vector field F(x,y,z) = (6y, 6x, z).

Show that F is a gradient vector field {eq}F = \Delta V {/eq} by determining the Function V which satisfies V(0,0,0) = 0.

## Vector Field:

In this problem, we'll use the gradient theorem:

{eq}\mathbf F= \nabla V= \dfrac{\partial V}{ \partial x } \mathbf i+\dfrac{\partial V}{ \partial y } \mathbf j+\dfrac{\partial V}{ \partial z } \mathbf k {/eq}.

To solve this problem, we need to integrate in terms of {eq}x,y,z {/eq} and use the gradient theorem to get the desired solution.

## Answer and Explanation:

We are given:

{eq}\mathbf F(x,y,z) = (6y, 6x, z) {/eq}

We need to find the scalar potential.

Since {eq}\mathbf F= \nabla V = \dfrac{\partial V}{...

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