Curl and Divergence

1. The definition of curl

$$curl \vec F=\nabla\times\vec F=\det
\begin{pmatrix}
\vec i&\vec j&\vec k\\
\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\
P&Q&R\end{pmatrix}$$

2. If the vector field is a gradient of another function, then its curl is 0.

$$curl(\nabla\vec F)=\vec 0$$

3. In a simple connected region, the curl of the vector field is 0 anywhere, if and only if the vector field is conservative.

$$curl \vec F=\vec 0 \iff \vec F \ is \ conservative$$

4. The definition of divergence

$$div \vec F =\nabla \cdot \vec F=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}$$

5.

$$div(curl\vec F)=0$$

6. Stokes’ theorem. A loop integral equals to the integral of its curl over the surface that is bounded by the loop. The direction of the loop and the surface normal vector are determined by the right-hand rule.

$$\oint_C \vec F\cdot d\vec r=\iint_S curl \vec F\cdot d\vec S$$

7. Divergence theorem. The flux through a closed surface equals to the integral of its divergence over the volume inside the surface

$$\iint_S \vec F\cdot d\vec S=\iiint_E div \vec FdV$$

8. If \(f\) is a function, \(\vec F\) is a vector field.

$$div(f\vec F)=\nabla f\cdot \vec F+f\ div\vec F$$