S e \mathbf{F}(\mathrm{x}, \mathrm{y}, \mathrm{z})=f_{1}(x, y, z) \boldsymbol{i}+f_{2}(x, y, z) \boldsymbol{j}+f_{3}(x, y, z) \boldsymbol{k} e \boldsymbol{\nabla}=\frac{\partial}{\partial x} \boldsymbol{i}+\frac{\partial}{\partial y} \boldsymbol{j}+\frac{\partial}{\partial z} \boldsymbol{k}
Mostre que \boldsymbol{\nabla} \times \mathbf{F}=\left(\frac{\partial \mathrm{f}_{3}}{\partial \mathrm{y}}-\frac{\partial \mathrm{f}_{2}}{\partial \mathrm{z}}\right) \mathbf{i}+\left(\frac{\partial \mathrm{f}_{1}}{\partial \mathrm{z}}-\frac{\partial \mathrm{f}_{3}}{\partial \mathrm{x}}\right) \mathbf{j}+\left(\frac{\partial \mathrm{f}_{2}}{\partial \mathrm{x}}-\frac{\partial \mathrm{f}_{1}}{\partial \mathrm{y}}\right) \mathbf{k} (Obs: o presente produto vetorial corresponde ao rotacional de um campo vetorial)
Passo 1
O enunciado pede para mostrarmos que, dado o campo vetorial