Gradient Operator Distributes over Addition

Theorem

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

Let $\map f {x_1, x_2, \ldots, x_n}, \map g {x_1, x_2, \ldots, x_n}: \mathbf V \to \R$ be differentiable real-valued functions on $\mathbf V$.

Let $\nabla f$ denote the gradient of $f$.

Then:

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

$\ds \nabla \paren {f + g}$

$=$

$\ds \sum_{k \mathop = 1}^n \frac {\partial \paren {f + g} } {\partial x_k} \mathbf e_k$

$\ds $

$=$

$\ds \sum_{k \mathop = 1}^n \paren {\frac {\partial f} {\partial x_k} \mathbf e_k + \frac {\partial g} {\partial x_k} \mathbf e_k}$

$\ds $

$=$

$\ds \sum_{k \mathop = 1}^n \frac {\partial f} {\partial x_k} \mathbf e_k + \sum_{k \mathop = 1}^n \frac {\partial g} {\partial x_k} \mathbf e_k$

$\ds $

$=$

$\ds \nabla f + \nabla g$

$\blacksquare$