Question

# Suppose that $\nabla \cdot \mathbf{F}=0$ and $\nabla \cdot \mathbf{G}=0$. Does $\mathbf{F}+\mathbf{G}$ necessarily have zero divergence?

Solution

Verified
Step 1
1 of 3

$\text{Let }\mathbf{F}=\left\text{ and }\mathbf{G}=\left$

Hence

$\operatorname{div}(\mathbf{F})=\left(\dfrac{\partial\ }{\partial x}\ f_1+\dfrac{\partial\ }{\partial y}\ f_2+\dfrac{\partial\ }{\partial z}\ f_3\right)\text{ and }\operatorname{div}(\mathbf{G})=\left(\dfrac{\partial\ }{\partial x}\ g_1+\dfrac{\partial\ }{\partial y}\ g_2+\dfrac{\partial\ }{\partial z}\ g_3\right)\qquad{(1)}$

From(1)

$\text{If }\operatorname{div}(\mathbf{F})=0\text{ and }\operatorname{div}(\mathbf{G})=0$

, we have

$\dfrac{\partial\ }{\partial x}\ f_1=\dfrac{\partial\ }{\partial y}\ f_2=\dfrac{\partial\ }{\partial z}\ f_3=\dfrac{\partial\ }{\partial x}\ g_1=\dfrac{\partial\ }{\partial y}\ g_2=\dfrac{\partial\ }{\partial z}\ g_3=0\qquad{(2)}$

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