Work done by F = (y, x) around the unit circle
∫C F·dr where F = (y, x, 0) along r(t) = (cos(t), sin(t), 0) equals 0 ≈ 0. A worked vector field line integral with an editable calculator.
∫C F·dr where F = (y, x, 0) = 0 ≈ 0. This field is conservative (it is the gradient of xy), so the work around any closed loop is zero.
Vector field F = (P, Q, R)
Each component in terms of x, y and z
Curve C: r(t) = (x(t), y(t), z(t))
Leave z(t) as 0 for a plane curve
The setup
∫C F·dr where F = (y, x, 0) = 0 ≈ 0
- Curve: r(t) = (cos(t), sin(t), 0) for t from 0 to 2*pi
- Arc length of the curve: 6.283185
- Type: vector field line integral
This field is conservative (it is the gradient of xy), so the work around any closed loop is zero.
More worked line integrals
Frequently asked questions
What is the value of this line integral? ▾
∫C F·dr where F = (y, x, 0) = 0 ≈ 0. This field is conservative (it is the gradient of xy), so the work around any closed loop is zero.
What curve is being integrated along? ▾
The curve is parametrised as r(t) = (cos(t), sin(t), 0) for t from 0 to 2*pi, and it has arc length 6.283185.
Does the direction of travel matter? ▾
Yes. Reversing the orientation of the curve flips the sign of a vector line integral, because the direction of dr reverses.
About this Calculator
∫C F·dr where F = (y, x, 0) along r(t) = (cos(t), sin(t), 0) equals 0 ≈ 0. A worked vector field line integral with an editable calculator.