Differential Operators
Table of Contents
- Vector field
- A vector field on \( \mathbb{R}^{n} \) is a function \( \mathbf{F}:X\mapsto \mathbb{R}^{n} \) where \( X\subset\mathbb{R}^{n} \)
- Scalar field
- A scalar field is a function \( f:X\mapsto\mathbb{R} \) where \( X\subset\mathbb{R}^{n} \)
- Gradient field (Conservative vector field)
- Is a vector field \( \mathbf{F}:X\mapsto\mathbb{R}^{n} \) which is the gradient of some function \( f:X\mapsto\mathbb{R} \) where \( X\subset\mathbb{R}^{n} \)
1. Del Operator
The del operator in \( \mathbb{R}^{n} \) is defined by
\[ \nabla = \sum_{i=1}^{n} \frac{\partial}{\partial x_{i}} \mathbf{e}_{i} \]
It maps a scalar field to a vector field.
For example, let \(f:\mathbb{R}^{3}\mapsto\mathbb{R}\), we have
\[ \nabla f = \frac{\partial}{\partial x}\mathbf{i}+\frac{\partial}{\partial y}\mathbf{j}+\frac{\partial}{\partial z}\mathbf{k} \]
2. Divergence
Let \( \mathbf{F}:X\mapsto\mathbb{R}^{n} \) be a differentiable vector field where \( X\subset\mathbb{R}^{n} \). The divergence of \( \mathbf{F} \) is defined by
\[ \mathop{\mathrm{div}} \mathbf{F} = \nabla \cdot \mathbf{F} = \sum_{i=1}^{n} \frac{\partial F_{i}}{\partial x_{i}} \]
3. Curl
Let \( \mathbf{F}:X\mapsto\mathbb{R}^{3} \) be a differentiable vector field where \( X\subset\mathbb{R}^{3} \). The curl of \( \mathbf{F} \) is defined by
\[ \mathop{\mathrm{curl}}\mathbf{F} = \nabla \times \mathbf{F} \]
Let \( f:X\mapsto \mathbb{R} \) be a scalar field of class \(C^{2}\) where \(X\subset\mathbb{R}^{3}\). Then
\[ \nabla \times (\nabla f) = \mathbf{0} \]
Let \( f(x_{1}, x_{2}, x_{3}) \), then \( \nabla f = (\frac{\partial f}{\partial x_{1}}, \frac{\partial f}{\partial x_{2}}, \frac{\partial f}{\partial x_{3}}) \). Therefore,
\[\begin{aligned} & \nabla \times (\nabla f) \\ = & \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x_{1}} & \frac{\partial}{\partial x_{2}} & \frac{\partial}{\partial x_{3}} \\ \frac{\partial f}{\partial x_{1}} & \frac{\partial f}{\partial x_{2}} & \frac{\partial f}{\partial x_{3}} \end{vmatrix} \\ = & (\frac{\partial^{2} f}{\partial x_{2}x_{3}} - \frac{\partial^{2} f}{\partial x_{3}x_{2}})\mathbf{i} - (\frac{\partial^{2} f}{\partial x_{1}x_{3}} - \frac{\partial^{2} f}{\partial x_{3}x_{1}})\mathbf{j} + (\frac{\partial^{2} f}{\partial x_{1}x_{2}} - \frac{\partial^{2} f}{\partial x_{2}x_{1}})\mathbf{k} \\ = & \mathbf{0} \end{aligned}\]
Let \( \mathbf{F}:X\mapsto\mathbb{R}^{3} \) be a vector field of class \( C^{2} \) where \(X\subset\mathbb{R}^{3}\). Then
\[ \nabla\cdot(\nabla\times \mathbf{F}) = 0 \]