Divergence Operator Distributes over Addition

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Theorem

Let $\map {\mathbf V} {x_1, x_2, \ldots, x_n}$ be a vector space of $n$ dimensions.

Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis of $\mathbf V$.


Let $\mathbf f$ and $\mathbf g: \mathbf V \to \mathbf V$ be vector-valued functions on $\mathbf V$:

$\mathbf f := \tuple {\map {f_1} {\mathbf x}, \map {f_2} {\mathbf x}, \ldots, \map {f_n} {\mathbf x} }$
$\mathbf g := \tuple {\map {g_1} {\mathbf x}, \map {g_2} {\mathbf x}, \ldots, \map {g_n} {\mathbf x} }$


Let $\nabla \cdot \mathbf f$ denote the divergence of $f$.


Then:

$\nabla \cdot \paren {\mathbf f + \mathbf g} = \nabla \cdot \mathbf f + \nabla \cdot \mathbf g$


Proof

\(\ds \nabla \cdot \paren {\mathbf f + \mathbf g}\) \(=\) \(\ds \sum_{k \mathop = 1}^n \frac {\map \partial {f_k + g_k} } {\partial x_k}\) Definition of Divergence Operator
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \paren {\frac {\partial f_k} {\partial x_k} + \frac {\partial g_k} {\partial x_k} }\) Linear Combination of Partial Derivatives
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \frac {\partial f_k} {\partial x_k} + \sum_{k \mathop = 1}^n \frac {\partial g_k} {\partial x_k}\)
\(\ds \) \(=\) \(\ds \nabla \cdot \mathbf f + \nabla \cdot \mathbf g\)

$\blacksquare$


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