intergrals by green’s theoreom in matlab

To compute a line integral using Green's Theorem in MATLAB, you need three inputs: two functions representing the vector field and one function representing the curve.

Here's an example code to calculate the line integral of a vector field F over a closed curve C using Green's Theorem:

main.m
syms x y;           % Define symbolic variables
F = [x^2+2*y, x-y]; % Define the vector field
P = F(1);           % Extract the x-component of the vector field
Q = F(2);           % Extract the y-component of the vector field
dPdy = diff(P,y);   % Calculate the partial derivative dP/dy
dQdx = diff(Q,x);   % Calculate the partial derivative dQ/dx
G = dPdy - dQdx;    % Define the function required for Green's Theorem
C = [-1 1 1 -1; 0 0 2 2]; % Define the curve as an array of points (x,y)
Area = polyarea(C(1,:),C(2,:)); % Calculate the area enclosed by the curve
Integral = double(integral2(G,-1,1,0,2)); % Use the double integral function to calculate the line integral
% The answer is the same as the area enclosed by the curve
fprintf("The integral of the vector field F over the curve C is %.2f \n",Integral);
fprintf("The area enclosed by the curve C is %.2f \n",Area);
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Explanation:

First, we define our vector field F(x,y). Then, we extract the x and y components P and Q respectively. Next, we find the partial derivatives of P and Q with respect to y and x respectively. We then find the function G as the difference between the two partial derivatives. We then define our curve C as an array of x and y coordinates. We find the area enclosed by the curve using the polyarea() function. Finally, we use the integral2() function to compute the line integral of F over C using Green's theorem. Since this example is over a closed curve, the answer is the same as the area enclosed by the curve.

Reference:

• Line Integrals Using MATLAB and Green's Theorem