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.

2. How Does the Calculator Work?

The calculator computes the circulation using numerical integration:

Where:

• \( \mathbf{F} \) — Vector field (e.g., "P(x,y), Q(x,y)")
• \( \mathbf{r}(t) \) — Parametric path (e.g., "x(t), y(t)")
• \( a \) to \( b \) — Parameter range

3. Importance of Circulation Calculation

Details: Circulation is fundamental in fluid dynamics, electromagnetism, and other physics applications. It helps determine if a field is conservative and relates to curl via Stokes' theorem.

4. Using the Calculator

Tips:

• Enter vector field components separated by commas (e.g., "y, -x")
• Enter parametric equations for the path (e.g., "cos(t), sin(t)")
• Specify the parameter range (typically 0 to 2π for circles)

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between circulation and flux?
A: Circulation is the line integral of a vector field along a closed path, while flux is the surface integral through a closed surface.

Q2: When is circulation zero?
A: Circulation is zero for conservative fields or when integrating around paths in irrotational regions.

Q3: How does this relate to Green's Theorem?
A: Green's Theorem connects circulation around a closed curve to the double integral of curl over the enclosed region.

Q4: Can I use this for 3D fields?
A: This calculator handles 2D fields. For 3D, you would need to use Stokes' Theorem.

Q5: What are common applications?
A: Calculating work done by force fields, analyzing fluid flow, and studying electromagnetic fields.