Line integral: definitions and properties, dependence of the integral on its path. Green’s theorem

Line integral: definitions and properties, dependence of the

integral on its path. Green’s theorem.

Green’s Theorem Statement

Let C be the positively oriented, smooth, and simple closed curve in a plane, and D be the region bounded by the C. If L and M are the functions of (x, y) defined on the open region, containing D and have continuous partial derivatives, then the Green’s theorem is stated as
Where the path integral is traversed counterclockwise along with C.

Green’s Theorem Proof

The proof of Green’s theorem is given here. As per the statement, L and M are the functions of (x, y) defined on the open region, containing D and having continuous partial derivatives. So based on this we need to prove:

Green’s Theorem Area

With the help of Green’s theorem, it is possible to find the area of the closed curves.
From Green’s theorem,
If in the formula,
= 1, then we have,
Therefore, the line integral defined by Green’s theorem gives the area of the closed curve. Therefore, we can write the area formulas as:

Green Gauss Theorem

If Σ is the surface Z which is equal to the function f(x, y) over the region R and the Σ lies in V, then

• 
  exists.
  
• 
  

It reduces the surface integral to an ordinary double integral.
Green’s Gauss theorem can be stated from the above expression.
If P(x, y, z), Q(x, y, z), and R((x, y, z) are the three points on V, and it is bounded by the region
and α, β, and γ are the direction angles, then

Green’s Theorem Example

Let us solve an example based on Green’s theorem.

Green’s Theorem Applications

Green’s Theorem is the particular case of Stokes Theorem in which the surface lies entirely in the plane. But with simpler forms. Particularly in a vector field in the plane. Also, it is used to calculate the area; the tangent vector to the boundary is rotated 90° in a clockwise direction to become the outward-pointing normal vector to derive Green’s Theorem’s divergence form.