Find the divergence theorem value for the function given by (ez , sin x, y2 )

To apply the divergence theorem, we need to find the divergence of the vector field (mathbf{F} = (e^z, sin x, y^2)).

The divergence of a vector field (mathbf{F} = (F_1, F_2, F_3)) is given by the formula:

[

nabla cdot mathbf{F} = frac{partial F_1}{partial x} + frac{partial F_2}{partial y} + frac{partial F_3}{partial z}

]

For (mathbf{F} = (e^z, sin x, y^2)):

1. Calculate (frac{partial F_1}{partial x} = frac{partial (e^z)}{partial x} = 0)
2. Calculate (frac{partial F_2}{partial y} = frac{partial (sin x)}{partial y} = 0)
3. Calculate (frac{partial F_3}{partial z} = frac{partial (y^2)}{partial z} = 0)

Combining these results, we find:

[

nabla cdot mathbf{F} = 0 + 0 + 0 = 0

]

Since the divergence of (mathbf{F}) is (0),