Question: Divergencia y rotacional
Encuentra la divergencia y el rotacional de los siguientes campos vectoriales:
F(x, y, z)=xyz i −x^2 y k
1. **Planteamiento del problema:**
Queremos encontrar la divergencia y el rotacional del campo vectorial $$\mathbf{F}(x,y,z) = xyz\mathbf{i} - x^2 y \mathbf{k}$$.
2. **Recordatorio de fórmulas:**
- 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 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. **Identificación de componentes:**
$$P = xyz, \quad Q = 0, \quad R = -x^2 y$$.
4. **Cálculo de la divergencia:**
$$\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(xyz) + \frac{\partial}{\partial y}(0) + \frac{\partial}{\partial z}(-x^2 y)$$
Calculamos cada derivada parcial:
- $$\frac{\partial}{\partial x}(xyz) = yz$$
- $$\frac{\partial}{\partial y}(0) = 0$$
- $$\frac{\partial}{\partial z}(-x^2 y) = 0$$
Por lo tanto:
$$\nabla \cdot \mathbf{F} = yz + 0 + 0 = yz$$.
5. **Cálculo del rotacional:**
Calculamos cada componente:
- Componente $$i$$:
$$\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = \frac{\partial}{\partial y}(-x^2 y) - \frac{\partial}{\partial z}(0) = -x^2 - 0 = -x^2$$
- Componente $$j$$:
$$- \left( \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} \right) = - \left( \frac{\partial}{\partial x}(-x^2 y) - \frac{\partial}{\partial z}(xyz) \right) = - \left( -2xy - xy \right) = -(-2xy - xy) = 3xy$$
- Componente $$k$$:
$$\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = \frac{\partial}{\partial x}(0) - \frac{\partial}{\partial y}(xyz) = 0 - xz = -xz$$
6. **Resultado final:**
- Divergencia:
$$\boxed{\nabla \cdot \mathbf{F} = yz}$$
- Rotacional:
$$\boxed{\nabla \times \mathbf{F} = -x^2 \mathbf{i} + 3xy \mathbf{j} - xz \mathbf{k}}$$