Theory

Vector fields and Green's theorem

Vector fields in the plane and in space, work along a curve, conservative fields and potential, irrotational fields and simply connected domains, Green's theorem and computation of areas.

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Vector fields
Work and circulation

Vector fields

Vector field

A vector field on an open set $A \subseteq R^{2}$ is a function $F : A \to R^{2}$ that assigns a vector to each point,

F (x, y) = (F_{1} (x, y), F_{2} (x, y)),

where $F_{1}$ and $F_{2}$ , the components of the field, are scalar functions on $A$ . In space a vector field on $A \subseteq R^{3}$ has three components, $F = (F_{1}, F_{2}, F_{3})$ . The field is said to be of class $C^{k}$ if its components are of class $C^{k}$ . A vector field is the mathematical model of a force that depends on the point, such as the gravitational field or an electric field; the typical picture shows at each point of the plane an arrow with the direction and the length of the associated vector.

Example

$F (x, y) = (- y, x)$ is a vector field of class $C^{\infty}$ on $R^{2}$ . At each point the vector $(- y, x)$ is perpendicular to the position vector $(x, y)$ , since the scalar product $(- y) \cdot x + x \cdot y$ is zero, hence the arrows rotate around the origin counterclockwise. An example in space is the field $F (x, y, z) = (y z, x z, x y)$ , of class $C^{\infty}$ on $R^{3}$ .

The field

F (x, y) = (- y, x)

at some points of three circles centered at the origin: the arrows rotate counterclockwise and get longer moving away from the origin

Remark

Every scalar function $f$ of class $C^{1}$ on an open set $A$ generates a vector field, its gradient $\nabla f = (f_{x}, f_{y})$ . The central question of this chapter is the inverse one: given a field $F$ , determine whether there exists a scalar function of which $F$ is the gradient. For the gradient we refer to the chapter on differential calculus.

Work and circulation

The line integral of the second kind of a field $F$ along an oriented curve $γ$ is called the work of $F$ along $γ$ . In the plane it is also written in the form

\int_{γ} F_{1} d x + F_{2} d y,

which in the computation becomes $\int_{a}^{b} (F_{1} (r (t)) x^{'} (t) + F_{2} (r (t)) y^{'} (t)) d t$ . If the curve is closed, that is, $r (a) = r (b)$ , the work is called the circulation of $F$ along $γ$ and is denoted by

\oint_{γ} F \cdot d r .

The name comes from physics: if $F$ is a force, the integral measures the work done by the force on a material point traveling along the curve. Circulation is the tool with which non-conservative fields are recognized, as the characterization theorem shows.

Computation of a work

Work of $F (x, y) = (y, x^{2})$ along the segment from $(0, 0)$ to $(1, 1)$ , parametrized by $r (t) = (t, t)$ with $t \in [0, 1]$ . Since $r^{'} (t) = (1, 1)$ and $F (r (t)) = (t, t^{2})$ , the work is

\int_{0}^{1} (t \cdot 1 + t^{2} \cdot 1) d t = \frac{1}{2} + \frac{1}{3} = \frac{5}{6} .

The theory continues with 7 more sections (conservative fields and potential, path independence, irrotational fields…), plus all worked examples, the solutions and the PDF.

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