韦达定理是一个重要的多元微积分工具，常见的五种公式如下：

1. 标量形式：$$\nabla f = \frac{\partial f}{\partial x} i + \frac{\partial f}{\partial y} j + \frac{\partial f}{\partial z} k$$

2. 散度形式：$$\nabla \cdot F = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$$

3. 旋度形式：$$\nabla \times F = \begin{vmatrix}i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix}$$

4. 二重积分形式：$$\int_{\partial D} F \cdot \mathrm{d}r = \iint_D \nabla \times F \cdot \hat{n} \, \mathrm{d}S$$

5. 线积分形式：$$\int_C F \cdot \mathrm{d}r = \iint_S (\nabla \times F) \cdot \hat{n} \, \mathrm{d}S$$