Line integrals

Level UniversityDifficulty ★★★★★Concept⌖ Open in the map

What is it?

Integrating a field along a curve: the work $\int_C F\cdot\dd r$ done by a force along a path. For gradient fields it depends only on the endpoints — the definition of a conservative force.

Formulas

\int_C F\cdot\dd r = \int_a^b F\big(r(t)\big)\cdot r'(t)\,\dd t, \qquad \int_C \nabla\phi\cdot\dd r = \phi(B) - \phi(A)

Where it shows up in computing

Classical mechanics★★★★★frequentPhysics and simulation
Work and potential energy; energy conservation for conservative forces.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

⚛ Physics and simulation

Green's theorem→Stokes' theorem→Electromagnetism (Maxwell's equations)★★★★★
Classical mechanics★★★★★
Classical mechanics→Physics engines★★★★★
Classical mechanics→Physics engines→N-body gravitational simulation★★★★★
Classical mechanics→Physics engines→Fluid dynamics and CFD★★★★★
Classical mechanics→Physics engines→Fluid dynamics and CFD→Weather and climate modelling★★★★★

3D Computer graphics

Green's theorem→Mesh processing (discrete differential geometry)★★★★★

What depends on it

Green's theorem

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.