Circulation

The circulation integral is a line integral or surface integral where the integrand is:

$$\vec{F}\cdot \vec{T}$$

where $\vec{F}$ is a vector field taken with the dot product of the tangent vector. The orientation matters when computing circulation integrals, because the unit tangent vector depends on the orientation.

For an open contour (partial circulation) in either ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$:

$${\int }_a^b\vec{F}\cdot \vec{T}dr$$

And for a closed contour:

$$\oint \vec{F}\cdot \vec{T}ds$$

Geometrically we think about circulation as the total work done by the vector field to move an object on a curve.

Computations

Note that $\vec{T}={\vec{r}}^{\prime }/\Vert r^{\prime }\Vert$, then

$$\vec{T}ds=d\vec{r}=⟨x^{\prime }(t),y^{\prime }(t)⟩dt$$ $$\oint \vec{F}\cdot \vec{T}ds={\int }_a^b\vec{F}\cdot {\vec{r}}^{\prime }dt$$

We always parameterise $x,y$ in terms of whatever makes up $r$. Note also that the following are equivalent:

$${\oint }_C\vec{F}\cdot \vec{T}ds={\oint }_{C}f(x,y(x))dx+g(x(y),y)dy$$

Alternate methods

For a vector field $\vec{F}=⟨x,y⟩$, if we re-write as a complex-valued function $f=x+iy$ and evaluate the same contour integral, then the real part of the result is the circulation.

See also

• Flux
• Stokes’ theorem