Find the divergence of the vector F= xe-x i + y j – xz k

To find the divergence of the vector field (mathbf{F} = x e^{-x} mathbf{i} + y mathbf{j} – xz mathbf{k}), we apply the divergence operator, which is given by:

[

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

]

where (mathbf{F} = F_1 mathbf{i} + F_2 mathbf{j} + F_3 mathbf{k}).

1. Identify the components:

– (F_1 = x e^{-x})

– (F_2 = y)

– (F_3 = -xz)

2. Calculate each partial derivative:

– (frac{partial F_1}{partial x} = frac{partial}{partial x}(x e^{-x}) = e^{-x} – x e^{-x} = (1 – x)e^{-x})

– (frac{partial F_2}{partial y} = frac{partial}{partial y}(y) = 1)

– (frac{partial F_3}{partial z} = frac{partial}{partial z}(-xz) = -x)

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