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)
- 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.
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
\[ \mathop{\mathrm{curl}}\mathbf{F} = \nabla \times \mathbf{F} \]
Proposition 1. Let \( f:X\mapsto \mathbb{R} \) be a scalar field of class \(C^{2}\) where \(X\subset\mathbb{R}^{3}\). Then
\[ \nabla \times (\nabla f) = \mathbf{0} \]
Proposition 2. Let \( \mathbf{F}:X\mapsto\mathbb{R}^{3} \) be a vector field of class \( C^{2} \) where \(X\subset\mathbb{R}^{3}\). Then
\[ \nabla\cdot(\nabla\times \mathbf{F}) = 0 \]