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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^n \frac {\partial f_k} {\partial x_k} + \sum_{k \mathop = 1}^n \frac {\partial g_k} {\partial x_k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \nabla \cdot \mathbf f + \nabla \cdot \mathbf g\) |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: Miscellaneous Formulas involving $\nabla$: $22.36$