The algebra rules for gradients Given a constant k and the gradients
\nabla f=\frac{\partial f}{\partial x} \mathbf{i}+\frac{\partial f}{\partial y} \mathbf{j}+\frac{\partial f}{\partial z} \mathbf{k}
and
\nabla g=\frac{\partial g}{\partial x} \mathbf{i}+\frac{\partial g}{\partial y} \mathbf{j}+\frac{\partial g}{\partial z} \mathbf{k}
use the scalar equations
\begin{array}{cc} \frac{\partial}{\partial x}(k f)=k \frac{\partial f}{\partial x}, & \frac{\partial}{\partial x}(f \pm g)=\frac{\partial f}{\partial x} \pm \frac{\partial g}{\partial x} \\\ \frac{\partial}{\partial x}(f g)=f \frac{\partial g}{\partial x}+g \frac{\partial f}{\partial x}, & \frac{\partial}{\partial x}\left(\frac{f}{g}\right)=\frac{g \frac{\partial f}{\partial x}-f \frac{\partial g}{\partial x}}{g^{2}} \end{array}
and so on, to establish the following rules.
\nabla(f+g)=\nabla f+\nabla g

Passo 1

O enunciado pede para provar a regra de soma do gradiente, ou seja,
\nabla(f+g)=\nabla f+\nabla g,
sabendo que
\nabla h =\frac{\partial h}{\partial x}\,\mathbf{i} +\frac{\partial h}{\partial y}\,\mathbf{j} +\frac{\partial h}{\partial z}\,\mathbf{k},
e usando a linearidade das derivadas parciais:
\frac{\partial}{\partial x}(f\pm g) =\frac{\partial f}{\partial x}\pm\frac{\partial g}{\partial x},
e análogas para y e z.

Passo 2

Aplicamos a propriedade de soma das derivadas parciais em cada coordenada:
\frac{\partial}{\partial x}(f+g) =\frac{\partial f}{\partial x}+\frac{\partial g}{\partial x},
\frac{\partial}{\partial y}(f+g) =\frac{\partial f}{\partial y}+\frac{\partial g}{\partial y},
\frac{\partial}{\partial z}(f+g) =\frac{\partial f}{\partial z}+\frac{\partial g}{\partial z}.

Passo 3

Reescrevendo em forma vetorial, temos
\nabla(f+g) =\Bigl(\tfrac{\partial f}{\partial x}+\tfrac{\partial g}{\partial x}\Bigr)\mathbf{i} +\Bigl(\tfrac{\partial f}{\partial y}+\tfrac{\partial g}{\partial y}\Bigr)\mathbf{j} +\Bigl(\tfrac{\partial f}{\partial z}+\tfrac{\partial g}{\partial z}\Bigr)\mathbf{k}.
Isso se decompõe em
\nabla(f+g) =\bigl(\tfrac{\partial f}{\partial x}\mathbf{i}+\tfrac{\partial f}{\partial y}\mathbf{j}+\tfrac{\partial f}{\partial z}\mathbf{k}\bigr) +\bigl(\tfrac{\partial g}{\partial x}\mathbf{i}+\tfrac{\partial g}{\partial y}\mathbf{j}+\tfrac{\partial g}{\partial z}\mathbf{k}\bigr) =\nabla f+\nabla g.

Resposta

\nabla(f+g)=\nabla f+\nabla g.

Exercícios relacionados