Question

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

Solution

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Let F=<f1,f2,f3> and G=<g1,g2,g3>\text{Let }\mathbf{F}=\left<f_1,f_2,f_3\right>\text{ and }\mathbf{G}=\left<g_1,g_2,g_3\right>

Hence

div(F)=( x f1+ y f2+ z f3) and div(G)=( x g1+ y g2+ z g3)(1)\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)

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

, we have

 x f1= y f2= z f3= x g1= y g2= z g3=0(2)\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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