Introduction to the Green’s Theorem

Green's Theorem is a fundamental concept in vector calculus that relates a line integral around a simple closed curve to a double integral over the plane region bounded by the curve. It is used to create a powerful connection between line integrals and area calculations. Let’s discuss Green's theorem, proof, formula, and its applications in detail.

Understanding of the Green’s Theorem

It is a fundamental theorem of calculus which involves the study of a relation between a line integral and double integral of a plane. It is used to integrate the derivatives in a particular plane. It is named after the British mathematician George Green, who first formulated it in the 19th century. To understand Green’s theorem, it is important to understand the concept of line integral and double integral. Let’s understand the concept of line integral and double integral first. 

Line Integral 

When a function is integrated along a curve, the integral is known as a line integral. We can integrate a vector-valued or a scalar function by using the line integral. Mathematically, the line integral formula can be written as;

$\int_C f(r) ds= \inte^b_a f[r(t)] r’(t) dt$

Double integral

When a function is integrated twice, it is known as the double integral. It is also known as the surface integral because it is used to integrate a function over a surface. Mathematically, the formula of the surface integral or double integral is;

$\int \int_S f(x,y,z) dS=\int \int_D f(\vec{r}(u,v))||\vec{r}_u\times \vec{r}_v||dA$

What is Green’s Theorem?

The statement of Green’s theorem is:

Let C be a closed smooth curve in a plane, D be the region bounded by the curve, and if F=(L, M) is the vector field defined in the region containing D such that L and M have continuous partial derivatives. Then, 

$∮_C (F.dr)=\int \int_D \left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right)dA$

Where, 

• $∮_C$ represents the line integral along the curve C.
• $(\frac{∂Q}{∂x} -\frac{∂P}{∂y} )$ represents the curl.
• $dA$ is the infinitesimal area element.

This theorem is used in physics as it has many applications related to fluid flow and many other concepts. Let's understand the proof of Green's theorem.

Proof of Green’s Theorem

To prove Green’s theorem, we start with a vector field F=(L, M) and C be a smooth curve enclosing a region D. We need to prove that,

$∮_C (F.dr)=\int \int_D \left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right)dA$

First, divide region D into small rectangular area elements. Suppose that the width and height of each area element are Δx and Δy respectively and the area of each rectangle is,

$\Delta A=\Delta x . \Delta y$

Applying the divergence theorem to each rectangular area element, we have

$∬_D\left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right) = ∬_D div(F) dA$

where div(F) represents the divergence of the vector field F.

Now, using the definition of divergence we can rewrite the right-hand side of the equation as a double integral over D:

$∬_D div(F) dA = ∬_D \left(\frac{\partial M}{\partial x}+\frac{\partial L}{\partial y}\right) dA$

Expanding this double integral, we have:

$∬_D \left(\frac{\partial M}{\partial x}+\frac{\partial L}{\partial y}\right) dA = ∬_D \left(\frac{\partial M}{\partial x}dA+\frac{\partial L}{\partial y}dA\right) $

Next, we will simplify the right side. First start with;

$∬_D\frac{∂M}{∂x} dA$

Using the fundamental theorem of calculus, we can rewrite the above integral as a line integral along the boundary of D. Since $\Delta A=\Delta x. \Delta y$,

$∬_D\frac{∂M}{∂x}\partal x\partial y = ∮ M dx$

Similarly, for the second term, we have:

$∬_D\frac{∂L}{∂x}\partal x\partial y = ∮ L dy$

Now, combining the two terms, we get:

$∬_D \left(\frac{\partial M}{\partial x}dA+\frac{\partial L}{\partial y}dA\right) = ∮ M dx + ∮ L dy

This is the line integral of the vector field F = (P, Q) along the curve C.

Finally, we have shown that:

∮_C F · dr = ∬_D \left(\frac{∂Q}{∂x} -\frac{∂P}{∂y}\right) dA$

Hence we have proved Green's theorem.

Applications of Green’s Theorem

Since Green’s Theorem is one of the fundamental theorems of calculus, it has many applications in physics, engineering, and mathematics. Some of the important applications of this theorem are:

• It allows us to analyze the behavior of vector fields within a region enclosed by a curve which is useful in fluid dynamics and electromagnetism where the flux and circulation of a field are important factors.
• This theorem can be used to calculate the work done by a field force along a closed path or curve. 
• It can be used to calculate the area of a region by using a line integral.
• It is one of the most appropriate ways to solve partial differential equations. 

The above are just a few examples from the wide range of Green’s theorem applications.

Conclusion

Green’s theorem is one of the important fundamental theorems of calculus which is used to relate a line integral to a double integral. It has a wide range of applications in mathematics, physics, engineering, and other branches of science. We can calculate the area under a curve, work done by a field force, and the solution of partial differential equations by using this theorem.