Flying a Loop Through the Wind

A drone flies a closed loop while a force field pushes on it. If the field is conservative, like gravity, the work it does around any closed loop is exactly zero. Add swirling wind and the loop costs energy: the line integral measures how much.

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How it works

Line integrals and conservative fields

In one sentence

For a vector field $\mathbf F$ and an oriented curve $C$ , $\int_C \mathbf F\cdot d\mathbf r = \int_a^b \mathbf F(\mathbf r(t))\cdot\mathbf r'(t)\,dt$ ; if $\mathbf F = \nabla f$ then $\int_C \mathbf F\cdot d\mathbf r = f(\mathbf r(b)) - f(\mathbf r(a))$ , which is 0 around any closed curve.

General formula

{\htmlData{sym=0}{W}} = {\htmlData{sym=1}{\oint_C}} {\htmlData{sym=2}{\mathbf F}}\cdot {\htmlData{sym=3}{d\mathbf r}}, \qquad {\htmlData{sym=2}{\mathbf F}} = \big(-{\htmlData{sym=4}{k}}x - {\htmlData{sym=5}{w}}y,\; -{\htmlData{sym=4}{k}}y + {\htmlData{sym=5}{w}}x\big)

Each step contributes force times distance in the direction you move. With a conservative force, uphill steps give back exactly what downhill steps took, so a round trip nets zero.

What the symbols mean

$W$
Work
Energy the field gives the drone around the loop (negative: energy the drone must spend).
$\oint_C$
Around the closed loop
Add up along the whole loop C, counterclockwise.
$\mathbf F$
Force field
The push on the drone at each point: pull toward base plus wind.
$d\mathbf r$
Path step
A tiny step along the loop.
$k$
Pull strength
The conservative part, like a spring tether to the base.
$w$
Wind swirl
The rotating wind: the non-conservative part.

The line integral adds up $\mathbf F\cdot d\mathbf r$ , the force along the direction of motion times each small step of the path.
The $-k(x, y)$ part pulls toward the base and is conservative: $\mathbf F = -\nabla U$ with $U = \tfrac k2(x^2 + y^2)$ , so its work around any closed loop is 0.
The $w(-y, x)$ part is a swirling wind. It has curl $2w$ , so it is not conservative and its work around a loop is $2w$ times the enclosed area.
A field is conservative exactly when its line integrals depend only on the endpoints, which on a simply connected region means zero curl.

Learn the math

Vector FieldsCalc III · Lesson 39, Line Integrals Line Integrals of Vector FieldsCalc III · Lesson 41, Line Integrals The Fundamental Theorem for Line IntegralsCalc III · Lesson 42, Line Integrals Conservative Vector FieldsCalc III · Lesson 43, Line Integrals Ampère's LawE&M · Lesson 19, Magnetic Fields and Forces

Learn the math

Vector FieldsCalc III · Lesson 39, Line Integrals

Line Integrals of Vector FieldsCalc III · Lesson 41, Line Integrals

The Fundamental Theorem for Line IntegralsCalc III · Lesson 42, Line Integrals

Conservative Vector FieldsCalc III · Lesson 43, Line Integrals

Ampère's LawE&M · Lesson 19, Magnetic Fields and Forces