Divergence and Curl
The Essentials
Divergence is the measure of outflowing-ness of a function at a given point. If the vector function points more outwards than inwards then the divergence at that point will be positive. If it points more inwards than outwards it will be negative. If the same amount flows in as it does out, it will be zero.
\[
\text{div } \vec{F}=P_x+Q_y+R_z
\]
Curl is the measure of a vector field's spin at a given point:
\[
\text{curl } \vec{F}=\langle R_y-Q_z,P_z-R_x,Q_x-P_y\rangle
\]
This formula can be remembered by taking the cross product of the gradient operator and the vector function:
\[
\text{curl } \vec{F}=\nabla \times \vec{F}
\]
The divergence of the curl of a vector function is always zero. The curl of a function is 0 if and only if a function is conservative.
Practice
1) Find the divergence of F at the point (1, 1, -1):
\[
\vec{F}=\langle x^2z,x^2z,x^2+xyz+y^2z\rangle
\]
2) Find the curl of F:
\[
\vec{F}=\langle x^2z,x^2z,x^2z\rangle
\]
Solutions:
1) 0
2) \( \langle -x^2,x^2-2xz,2xz\rangle \)