1. What is Green's Theorem?

Green's Theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It's a special case of Stokes' Theorem in two dimensions.

2. How Does the Calculator Work?

The calculator uses the Green's Theorem formula:

Where:

• \( P \) — Function of x and y multiplied by dx in the line integral
• \( Q \) — Function of x and y multiplied by dy in the line integral
• \( C \) — Positively oriented, piecewise smooth, simple closed curve
• \( D \) — Region bounded by C

Explanation: The calculator numerically computes the double integral of (∂Q/∂x - ∂P/∂y) over the specified region.

3. Importance of Green's Theorem

Details: Green's Theorem is fundamental in vector calculus, connecting line integrals and double integrals. It's used in physics for calculating work, circulation, and flux.

4. Using the Calculator

Tips:

• Enter P and Q as functions of x and y (e.g., "x^2*y" or "sin(x)+cos(y)")
• Specify x bounds as numbers
• Specify y bounds as functions of x or constants
• The calculator uses numerical approximation with 1000 steps in each direction

5. Frequently Asked Questions (FAQ)

Q1: What's the relationship between Green's and Stokes' Theorems?
A: Green's Theorem is a special case of Stokes' Theorem restricted to two dimensions in the xy-plane.

Q2: What types of regions does Green's Theorem work for?
A: It works for regions bounded by simple, closed, piecewise-smooth curves.

Q3: How accurate is the numerical approximation?
A: Accuracy depends on step size. The calculator uses 1000 steps in each direction for reasonable accuracy.

Q4: Can I use this for non-rectangular regions?
A: Yes, by specifying y bounds as functions of x, you can handle various region shapes.

Q5: What are common applications of Green's Theorem?
A: Applications include calculating areas, work done by force fields, and fluid flow problems.