Green's and Stokes' Theorems

Green's and Stokes' Theorems

These powerful theorems connect line integrals and surface integrals, forming the foundation of vector calculus. Here's what you need to know:

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:

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

Where:

• $$P$$ and $$Q$$ are functions of $$(x,y)$$ with continuous partial derivatives
• $$C$$ is a positively oriented (counterclockwise), simple closed curve
• $$D$$ is the region enclosed by $$C$$

Key insight: Green's Theorem converts a line integral around a closed curve to a double integral over the region inside the curve, which is often easier to evaluate.

Example Application

Calculate $$\oint_C (x^2\,dy - y^2\,dx)$$ where $$C$$ is the unit circle centered at the origin.

Solution using Green's Theorem:

Here, $$P = -y^2$$ and $$Q = x^2$$

\[\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 2x - (-2y) = 2x + 2y\]

Converting to polar coordinates: $$x = r\cos\theta$$, $$y = r\sin\theta$$, $$dA = r\,dr\,d\theta$$

\[\iint_D (2x + 2y)\,dA = \int_0^{2\pi}\int_0^1 (2r\cos\theta + 2r\sin\theta)r\,dr\,d\theta\]

After integration: $$\oint_C (x^2\,dy - y^2\,dx) = 0$$

Stokes' Theorem

Stokes' Theorem extends Green's Theorem to three dimensions, relating a line integral around a closed curve C to a surface integral over any surface S bounded by C:

\[\oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot \vec{n}\,dS\]

Where:

• $$\vec{F}$$ is a vector field with continuous partial derivatives
• $$C$$ is a simple closed curve oriented positively
• $$S$$ is any surface bounded by $$C$$
• $$\vec{n}$$ is the unit normal vector to $$S$$
• $$\nabla \times \vec{F}$$ is the curl of the vector field

Key insight: Stokes' Theorem allows us to convert a difficult line integral into a potentially simpler surface integral, or vice versa.

Example Application

Calculate $$\oint_C (y\,dx + z\,dy + x\,dz)$$ where $$C$$ is the circle $$x^2 + y^2 = 1$$ in the $$xy$$-plane, oriented counterclockwise.

Solution using Stokes' Theorem:

Here, $$\vec{F} = (y, z, x)$$

\[\nabla \times \vec{F} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ y & z & x \end{vmatrix} = (1, 1, 1)\]

Since $$C$$ lies in the $$xy$$-plane, we can use the flat disk as our surface $$S$$, with normal vector $$\vec{n} = (0, 0, 1)$$

\[\iint_S (\nabla \times \vec{F}) \cdot \vec{n}\,dS = \iint_S 1\,dS = \text{Area of the unit circle} = \pi\]

Therefore, $$\oint_C (y\,dx + z\,dy + x\,dz) = \pi$$

Connection Between the Theorems

Green's Theorem is actually a special case of Stokes' Theorem when the vector field and curve lie in the $$xy$$-plane. Both theorems are special cases of the more general Generalized Stokes' Theorem, which unifies many fundamental theorems in vector calculus.

Applications: These theorems are essential in physics, particularly in electromagnetism, fluid dynamics, and thermodynamics. They allow us to convert between different forms of physical laws and simplify complex calculations.