1. What is Circulation in Vector Calculus?

Circulation measures the tendency of a vector field to rotate around a closed path. It's calculated as the line integral of a vector field around a closed curve, representing the "flow" of the field along the path.

2. How Does the Calculator Work?

The calculator uses the circulation formula:

Where:

• \( \mathbf{F} \) — Vector field (F_x, F_y)
• \( C \) — Closed path of integration
• \( d\mathbf{r} \) — Differential path element

Explanation: The circulation is computed by integrating the dot product of the vector field with the tangent vector along the path.

3. Importance of Circulation Calculation

Details: Circulation is fundamental in fluid dynamics, electromagnetism, and other physics applications. It helps determine rotational properties of vector fields and is related to curl through Stokes' theorem.

4. Using the Calculator

Tips: Enter the x and y components of your vector field as functions of x and y (e.g., "y" or "-x"). Select the path type (circle, ellipse, rectangle, or custom). For custom paths, additional parameterization may be required.

5. Frequently Asked Questions (FAQ)

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

Q2: How is circulation related to curl?
A: By Stokes' theorem, circulation around a loop equals the integral of the curl over the enclosed surface.

Q3: When is circulation zero?
A: Circulation is zero for conservative fields or when the path encloses no rotation in the field.

Q4: Can I compute circulation in 3D?
A: Yes, but this calculator currently handles 2D cases. 3D circulation would require parameterized paths in space.

Q5: What are common applications of circulation?
A: Used in aerodynamics (wing lift), electromagnetism (Faraday's law), and fluid dynamics (vortex strength).