Integrate fields along curves and over surfaces; used for work, flux and circulation.
A line integral ∫F·dr sums a field along a path (work); a surface integral ∬F·n dS gives flux through a surface. For conservative fields the line integral depends only on endpoints.
Work by gravity from height h₁ to h₂ is mg(h₁-h₂) regardless of path.
Line and surface integrals describe work along a path and flow through a surface, and they are the language of field theory.
Integrate fields along curves and over surfaces; used for work, flux and circulation.
Line and surface integrals describe work along a path and flow through a surface, and they are the language of field theory.
The work done along the path, which is the integral of the field component parallel to the path. It depends on the path only if the field is conservative.
Flux integrates the normal component over a surface. Circulation integrates the tangential component around a closed curve.
Typical applications include computing the work done by a force moving a robot along a curved path, calculating the flux of air through a duct cross-section and finding the length of a cable under a varying load.
Study "Coordinate systems" first (topic 44 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
Parameterise the curve or surface. Closed-loop integral measures circulation. Conservative if curl is zero.
Work by gravity from height h₁ to h₂ is mg(h₁-h₂) regardless of path.
The next topic is "Divergence and curl", which builds directly on this one.
It is classed as Intermediate in Engineering Mathematics (topic 45 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.
Want to practise? Open the interactive course for quick checks, quizzes, simulations and progress tracking.