Curl and Divergence

Definition

Assume that \( \overrightarrow F:A\to\mathbb R^3 \) is a vector field, where \( A\subseteq \mathbb R^3 \). Assume that \( \overrightarrow F=\langle P,Q,R\rangle \) where \( P \), \( Q \), and \( R \) are differentiable functions on \( A \). The curl of the vector field \( \overrightarrow F \) is defined as \[ \mbox{curl }\overrightarrow F=\left\langle\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right\rangle.\] The curl of the vector field is often denoted by \( \nabla \times \overrightarrow F \), as it can be taught of as a cross product of a formal vector \( \left\langle\frac{\partial }{\partial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z}\right\rangle \) with the vector \( \overrightarrow F \).

The divergence of the vector field \( \overrightarrow F \) is defined as \[ \mbox{div }\overrightarrow F= \frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}.\] The divergence is also denoted as \( \nabla \cdot \overrightarrow F \).

Example 1.
 

If \( f \) is a function and \( \overrightarrow{F} \) a vector field defined on a domain \( D\subseteq \mathbb R^3 \) prove the following equality: \[ \nabla \cdot \left(f\overrightarrow F\right)=\nabla f\cdot \overrightarrow F+f\nabla\cdot \overrightarrow{F}.\]

Example 2.
 

Assume that \( \overrightarrow F \) is a vector field whose components have continuous second partial derivatives. Prove that \( \mbox{div }\left(\mbox{curl }\overrightarrow F\right)=0 \).

Example 3.
 

Assume that \( f \) and \( g \) are two functions that have continuous second order partial derivatives. Prove that \[ \nabla \cdot \left(\nabla (fg)\right)= f \nabla \cdot (\nabla g)+ g\nabla\cdot (\nabla f)+2\nabla f\cdot \nabla g.\]


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