1. What is Circulation?

Circulation is a line integral of a vector field around a closed curve in the plane or in space. It measures the tendency of the field to circulate around the path.

2. How Does the Calculator Work?

The calculator uses the circulation formula:

Where:

• \( \mathbf{F} \) — Vector field (e.g., F = [P, Q, R])
• \( C \) — Closed path/curve
• \( d\mathbf{r} \) — Differential path element

Explanation: The integral sums the dot product of the vector field with the tangent vector along the path.

3. Importance of Circulation Calculation

Details: Circulation is important in fluid dynamics, electromagnetism, and other fields to measure rotational tendency or work done around a closed loop.

4. Using the Calculator

Tips: Enter the vector field components (comma-separated), parametric path equation, parameter variable, and integration limits.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between circulation and flux?
A: Circulation is a line integral (work around a path), while flux is a surface integral (flow through a surface).

Q2: When is circulation zero?
A: For conservative fields, circulation around any closed path is zero.

Q3: How does Stokes' theorem relate to circulation?
A: Stokes' theorem connects circulation to curl of the field over a surface bounded by the path.

Q4: What are common applications of circulation?
A: Used in aerodynamics (lift calculation), electromagnetism (Ampere's law), and fluid flow analysis.

Q5: Can circulation be negative?
A: Yes, negative circulation indicates flow in the opposite direction to the path orientation.