Divergencia Rotacional 29A2Ff

1. **Planteamiento del problema:** Encontrar la divergencia y el rotacional del campo vectorial $$\mathbf{F}(x,y,z) = x^2 y z \mathbf{i} + x y^2 z \mathbf{j} + x y z^2 \mathbf{k}$$. 2. **Fórmulas importantes:** - La divergencia de un campo vectorial $$\mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}$$ es $$\nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$$. - El rotacional de $$\mathbf{F}$$ es $$\nabla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} - \left( \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k}$$. 3. **Calcular la divergencia:** - $$P = x^2 y z$$, $$Q = x y^2 z$$, $$R = x y z^2$$. - $$\frac{\partial P}{\partial x} = \frac{\partial}{\partial x} (x^2 y z) = 2 x y z$$. - $$\frac{\partial Q}{\partial y} = \frac{\partial}{\partial y} (x y^2 z) = 2 x y z$$. - $$\frac{\partial R}{\partial z} = \frac{\partial}{\partial z} (x y z^2) = 2 x y z$$. - Por lo tanto, $$\nabla \cdot \mathbf{F} = 2 x y z + 2 x y z + 2 x y z = 6 x y z$$. 4. **Calcular el rotacional:** - $$\frac{\partial R}{\partial y} = \frac{\partial}{\partial y} (x y z^2) = x z^2$$. - $$\frac{\partial Q}{\partial z} = \frac{\partial}{\partial z} (x y^2 z) = x y^2$$. - $$\frac{\partial R}{\partial x} = \frac{\partial}{\partial x} (x y z^2) = y z^2$$. - $$\frac{\partial P}{\partial z} = \frac{\partial}{\partial z} (x^2 y z) = x^2 y$$. - $$\frac{\partial Q}{\partial x} = \frac{\partial}{\partial x} (x y^2 z) = y^2 z$$. - $$\frac{\partial P}{\partial y} = \frac{\partial}{\partial y} (x^2 y z) = x^2 z$$. - Componente $$\mathbf{i}$$: $$\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = x z^2 - x y^2 = x (z^2 - y^2)$$. - Componente $$\mathbf{j}$$: $$- \left( \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} \right) = - (y z^2 - x^2 y) = - y (z^2 - x^2)$$. - Componente $$\mathbf{k}$$: $$\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = y^2 z - x^2 z = z (y^2 - x^2)$$. - Por lo tanto, $$\nabla \times \mathbf{F} = x (z^2 - y^2) \mathbf{i} - y (z^2 - x^2) \mathbf{j} + z (y^2 - x^2) \mathbf{k}$$. 5. **Respuesta final:** - Divergencia: $$\boxed{6 x y z}$$. - Rotacional: $$\boxed{\mathbf{\nabla} \times \mathbf{F} = x (z^2 - y^2) \mathbf{i} - y (z^2 - x^2) \mathbf{j} + z (y^2 - x^2) \mathbf{k}}$$.