Flux Across a Plane Curve

Definition

Flux measures the net rate at which a vector field crosses a curve $C$ in a two-dimensional plane, calculated by integrating the field’s normal component. $\text{Flux} = \oint_C \mathbf{F} \cdot \mathbf{n} ds = \oint_C M dy - N dx$

• How to read: “The flux equals the closed line integral of the dot product of F and n with respect to s, which is equivalent to the integral of M dy minus N dx.”
• Meaning: Net rate the vector field $\mathbf{F} = M\mathbf{i} + N\mathbf{j}$ crosses curve C perpendicular to the boundary—throughput across a 2D closed contour.

Why It Matters

Flux is the fundamental measure of throughput in any transport system; whether calculating the rate of coolant flow in a reactor or airflow over a wing, understanding how fields cross boundaries is essential for managing energy and material exchange.

Core Concepts

• Normal Vector ($\mathbf{n}$): The unit vector perpendicular to the curve, usually chosen to point outward from a closed region.
• Throughput: Flux represents the “amount” of the field passing through the boundary per unit time.
• Scalar Form: For $\mathbf{F} = M\mathbf{i} + N\mathbf{j}$, the flux integral simplifies to the line integral of $M dy - N dx$.

Connected Concepts

• Phase Plane
• The Cycloid
• Gallery of Polar Curves