Labware - MA35 Multivariable Calculus - Two Variable Calculus



Vector Fields, Curl and Divergence


A vector field is defined by a function which associates a vector with each point in the domain of the function.

In two dimensions, a vector field V can be expressed in terms of the functions p(x,y) and q(x,y) as V(x,y) = (p(x,y),q(x,y)).

Note that for the gradient vector field, p = fx and q = fy.

The divergence of a vector field V(x,y) = (p(x,y),q(x,y)) is defined as div V = ∇ ⋅ V = px + qy.

The scalar curl of a two-dimensional vector field is defined as scalar curl V = -py(x,y)+qx(x,y).

If a two-dimensional vector field F(p,q) is conservative, then its curl is identically zero since py = qx, ∇ {times} F = (0,0,qx-py) = (0,0,0).