Scalar Field Line Integral Calculator

Definition

∫_C f(x,y) ds = ∫_a^b f(x(t),y(t)) √((dx/dt)² + (dy/dt)²) dt

Key Components

• Scalar field f(x,y)
• Path parameterization (x(t), y(t))
• Arc length element ds
• Parameter interval [a,b]

Common Parameterizations

• Line: r(t) = (1-t)P₁ + tP₂
• Circle: r(t) = (R cos(t), R sin(t))
• Parabola: r(t) = (t, at²)

Key Properties

• Path dependence
• Additivity over paths
• Orientation sensitivity
• Scalar multiplication

Physical Applications

• Work done by force field
• Mass of curved wire
• Heat flow along path
• Fluid flow resistance

Path Independence

• Conservative fields
• Potential functions
• Green's theorem conditions

Multiple Integration

• Change of variables
• Parametric curves
• Numerical methods