Question

What is the Gauss Green theorem?

Answer 1

The Gauss-Green Theorem (also known as Green's Theorem) is a fundamental result in vector calculus that relates a line integral around a simple, closed curve to a double integral over the region enclosed by the curve.

In simple terms, the theorem connects the circulation or flux of a vector field along the boundary of a region to the curl or divergence of the vector field within the region itself.

Statement of Green's Theorem:

If you have a vector field \( \mathbf{F} = (P(x, y), Q(x, y)) \), where \( P \) and \( Q \) are continuously differentiable functions, and \( C \) is a positively oriented, simple, closed curve enclosing a region \( D \), then Green's Theorem states:

\[
\oint_C \left( P(x, y) \, dx + Q(x, y) \, dy \right) = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA
\]

Where:

1. The left-hand side is a line integral around the curve \( C \).
   

1. The right-hand side is a double integral over the area \( D \) enclosed by \( C \).
   

1. \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \) represents the curl of the vector field.
   

Intuition:

1. The line integral on the left measures the total "circulation" of the vector field around the boundary curve.
   

1. The double integral on the right measures the total curl (or rotation) of the vector field inside the region.
   

Common Applications:

1. Fluid dynamics: to calculate the circulation of a fluid around a closed curve.
   

1. Electromagnetism: used in Maxwell's equations to relate electric and magnetic fields.
   

In essence, Green's Theorem helps convert a complicated integral around a curve into a potentially simpler integral over the region inside the curve.