1. What is Circulation in Vector Calculus?

Circulation measures the tendency of a vector field to rotate around a closed curve. It's calculated as the line integral of the vector field around the curve and is fundamental in fluid dynamics and electromagnetism.

2. How Does the Calculator Work?

The calculator uses the circulation formula:

Where:

• \( \mathbf{F} \) — Vector field (e.g., \( P\mathbf{i} + Q\mathbf{j} \))
• \( C \) — Closed curve (circle, ellipse, rectangle, etc.)
• \( d\mathbf{r} \) — Differential displacement vector along the curve

Explanation: The calculator numerically approximates the line integral around the specified closed curve.

3. Importance of Circulation Calculation

Details: Circulation is crucial in fluid dynamics (measuring vorticity) and electromagnetism (calculating work done by a force field). It's related to curl via Stokes' theorem.

4. Using the Calculator

Tips: Enter the vector field components separated by commas (e.g., "-y,x" for \( -y\mathbf{i} + x\mathbf{j} \)). Select curve type and provide parameters (radius for circle, semi-axes for ellipse, etc.).

5. Frequently Asked Questions (FAQ)

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

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

Q3: What are typical units for circulation?
A: Circulation is unitless when using normalized vector fields, but may have physical units in applications (m²/s for velocity fields).

Q4: When is circulation zero?
A: For conservative fields or when the curve encloses no rotation in the field.

Q5: Can I use this for 3D fields?
A: This calculator handles 2D fields. For 3D, you'd need to specify a closed 3D path.