Vector Fields and Line Integrals
Attach a vector to every point of a region, then integrate along a curve to accumulate work done by the field or mass along a wire.
What a learner can do afterwards
- Sketch a simple vector field and describe its flow
- Parameterise a curve and evaluate the work integral along it
- Distinguish integrating a scalar along a curve from integrating a field along it
1 · Read
A vector field pins one arrow to every point of a region. Think of a wind map: at each town the arrow shows which way the air moves and how fast. To sketch a field, pick a grid of points and draw each arrow starting at its point. Then look for structure: sources where arrows spread out, sinks where they crowd together, and swirls where they loop around. Long arrows mean fast flow there, and short ones mean slow drift.
Say you walk along a curving path through that wind. Dotting the field with your direction of travel at each step and adding it all up gives the work the field does on you. The drill is always the same: parametrise the curve as r(t), plug it into the field, dot with dr, and integrate from the start t to the end t. A scalar line integral is different: it multiplies a plain function by little bits of arc length ds, which gives things like the mass of a wire from its density.
Direction matters for fields but not for plain functions. Walk the same curve backwards and every tangent flips, so a vector line integral changes sign. The scalar version only cares about the function value and your speed, so it stays exactly the same. Remember the split: F dot dr uses the tangent direction, while f ds uses only speed.
Let the question words pick the tool for you. Work, circulation, or flow along a curve all ask for the vector form with F dot dr. Mass of a wire or average value along a curve asks for the scalar form with f ds.
A field puts an arrow at each point, and you add its push along a curve for work or a function along a curve for mass.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.