Labware - MA35 Multivariable Calculus - Two Variable Calculus
 MA35 Labs 2 » Two Variable Calculus Contents2.4 Integration 2.5 Vector Analysis 2.5.2 Path Integrals 2.5.4 Vector Fields, Curl and Divergence 2.5.6 Green's Theorem Search

Vector Fields, Curl and Divergence

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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).

Demos

 Divergence and Curl in 2D This demo shows, in the first window, the vector field V(x,y). In the "Divergence" window, the z-coordinate of the red surface indicates the value of the divergence at (x,y): div((x,y)) = f(x,y) = z. In the "Curl" window, the z-coordinate of the green surface indicates the value of the scalar curl at (x,y).