Vector fields

What is it?

Formulas

$$F : \R^3 \to \R^3, \qquad F(x,y,z) = \big(P, Q, R\big)$$

$$\dot x(t) = F\big(x(t)\big)$$

streamlines / particle tracing

Where it shows up in computing

• Fluid dynamics and CFD★★★★★fundamentalPhysics and simulation

  The state of a fluid simulation is a velocity field $u(x, t)$.

• Electromagnetism (Maxwell's equations)★★★★★fundamentalPhysics and simulation

  Electric and magnetic fields $E$ and $B$ are vector fields.

• Physics engines★★★★★frequentPhysics and simulation

  Wind, gravity wells and particle effects are applied as force fields.

Where is it used?

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

What depends on it

• Divergence
• Curl
• Line integrals
• Surface integrals and flux