Skip to content

Vector Fields

A vector field assigns a vector to every point in a region. In the plane, a vector field is usually written as
F(x,y)=P(x,y)i+Q(x,y)j.\begin{equation*}\mathbf{F}(x,y)=P(x,y)\mathbf{i}+Q(x,y)\mathbf{j}.\end{equation*}
The arrow at (x,y)(x,y) describes the direction and magnitude of the field at that point.
Vector fields appear in many parts of mathematics and physics. A fluid velocity field tells us how a fluid parcel moves, a gravitational field tells us the force on a mass, and electric or magnetic fields describe forces on charged particles.
The divergence of a two-dimensional vector field F=Pi+Qj\mathbf{F}=P\mathbf{i}+Q\mathbf{j} is
div⁡F=∂P∂x+∂Q∂y.\begin{equation*}\operatorname{div}\mathbf{F}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}.\end{equation*}
It measures the local tendency of the field to flow outward from a point. Positive divergence means the point behaves like a source, negative divergence means it behaves like a sink, and zero divergence means there is no local creation or destruction of flow.
For incompressible fluid flow, div⁡F=0\operatorname{div}\mathbf{F}=0 everywhere: locally, fluid volume is preserved.
For an incompressible fluid, the mathematical condition is
div⁡F=0.\begin{equation*}\operatorname{div}\mathbf{F}=0.\end{equation*}
The flow can bend, accelerate, and move around obstacles, but it does not appear or disappear inside the region.
A Joukowsky map transforms circular flow into airfoil-like flow.
The Joukowsky map
J(z)=z+1z\begin{equation*}J(z)=z+\frac{1}{z}\end{equation*}
transforms circles into shapes resembling airfoils. This gives a bridge between simple potential flow around a cylinder and more realistic-looking flow around a wing.
The curl of a two-dimensional vector field F=Pi+Qj\mathbf{F}=P\mathbf{i}+Q\mathbf{j} is the scalar
curl⁡F=∂Q∂x−∂P∂y.\begin{equation*}\operatorname{curl}\mathbf{F}=\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}.\end{equation*}
It measures the local tendency of the field to rotate around a point.
Positive curl corresponds to counterclockwise local rotation, while negative curl corresponds to clockwise local rotation.
The value of curl changes from point to point in a vector field.
Because curl is computed from partial derivatives, it is determined by how the field changes near a point, not by the whole picture at once. Moving the sample point through the same field can therefore produce positive, negative, or nearly zero curl.

References