[b]Circulation[/b]. Suppose that vector field [math]\textbf{F}[/math] represents the velocity field of a fluid through a region of space. In these circumstances, the integral of [math]\textbf{F\cdot T}[/math] along a curve in the region gives the fluid's flow along the curve. If the curve is closed, we call this the [i]circulation [/i]of field [math]\textbf{F}[/math] around the curve.[br] Circulation = [math]\int_C\textbf{F\cdot T}[/math][math]ds[/math].[br][b]Flux[/b]. We may wish to determine the rate at which fluid is entering or leaving a region enclosed by a smooth simple closed curve [math]C[/math] in the [math]xy[/math]-plane. To do this, we calculate the line integral over [math]C[/math] of [math]\textbf{F\cdot n}[/math], the scalar component of the fluid's velocity field in the direction of the curve's outward-pointing normal vector. The value of this integral is called the flux of [math]\textbf{F}[/math] across [math]C[/math].[br] Flux = [math]\int_C\textbf{F\cdot n}[/math][math]ds[/math].[br][br]Circulation is the integral of the tangential component of [math]\textbf{F}[/math]; flux is the integral of the normal component of [math]\textbf{F}[/math].[br][br]In the figure, click the button that says "Circulation" or "Flux" to toggle between the two calculations. Drag the slider to see the integral in action, and to identify where the integrand is positive or negative.

[i]Developed for use with Thomas' Calculus and [url=https://www.pearson.com/en-us/subject-catalog/p/interactive-calculus-early-transcendentals-single-variable/P200000009666]Interactive Calculus[/url], published by Pearson.[/i]