Module 42: Vector fields, scalar line integrals, work, and circulation

A. Prerequisites and observable goals#

B. The central idea: integrate along a path, not across an interval of space#

A scalar field assigns one number to each point, such as temperature or density. A vector field assigns a vector, such as force or fluid velocity. A path samples the field along a curve. Scalar line integrals weight length; vector line integrals weight directed displacement. They answer different questions even when both involve the same curve.

C. Two line integrals with different orientation behavior#

For a piecewise smooth parameterized curve $\vv r(t)$, $a\le t\le b$,

while

The first is unchanged by reversing a once-traced curve; the second changes sign. A nonnegative linear density makes the first a wire mass. A force field makes the second work, because only the force component along displacement contributes. Notation $P\dd x+Q\dd y+R\dd z$ denotes the same vector work integral for $\vv F=\langle P,Q,R\rangle$. [O3Line]

For a closed path, the work-type integral is also called circulation in a velocity-field context. In a counterclockwise oriented simple closed plane curve, the outward normal is to the right of travel. The planar outward flux is

This measures crossing of the boundary, not motion along it. The next module relates circulation and flux to area integrals.

D. Why work uses a dot product#

A small displacement $\Delta\vv r$ receives work approximately $\vv F\cdot\Delta\vv r$: a force perpendicular to the motion does no work, while a force opposed to motion contributes negatively. Summing and limiting gives the integral. Wire mass instead uses positive segment lengths, so it contains $\|\vv r'\|$, not a signed coordinate derivative.

A rotating field $\vv F=\langle-y,x\rangle$ is tangent to circles centered at the origin: it has circulation but zero normal flux through those circles. A radial field $\vv G=\langle x,y\rangle$ has positive outward flux but zero circular work. This distinction previews curl versus divergence.

E. A parameterization procedure#

State the curve and orientation, choose a parameter interval tracing it as intended, and differentiate the parameterization. Substitute every coordinate in the field. Then choose the final factor: speed for $\dd s$, tangent vector for $\dd\vv r$, or an oriented normal for flux. Compute and check units and signs. Repeated tracing counts repeatedly, so a geometric picture alone does not fix the integral without its parameter information.

F. Worked examples#

Work along a segment. For $\vv F=\langle y,x\rangle$ from $(0,0)$ to $(2,1)$, use $\vv r(t)=\langle2t,t\rangle$, $0\le t\le1$. Then $\vv F(\vv r)=\langle t,2t\rangle$ and $\vv r'=\langle2,1\rangle$, giving work $\int_0^1(2t+2t)\dd t=2$.

Flux versus circulation. On the counterclockwise unit circle, $\vv r=(\cos t,\sin t)$, $\vv r'=(-\sin t,\cos t)$, and outward normal $\vv n=(\cos t,\sin t)$. For the radial field $\vv G=\vv r$, the work integrand is zero while flux integrand is one. Total circulation is zero; outward flux is $2\pi$.

G. Independent practice#

F foundation C standard course work T transfer S optional stretch L runnable lab or project. Attempt the complete set before opening its solutions.

H. Complete solutions: Module 42#

Compare the first step where your work differs. A corrected answer is not yet an independent success.

Open every solution

I. Retention and readiness#

Retrieve M31.05 and M34.02. Say whether the problem is asking for density times length, force along motion, circulation, or flux across a boundary. Advance when you can predict orientation behavior before performing the integral.