File:Surface integral illustration.png

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Description	العربية: تعريف التكامل السطحي يعتمد على تقسيم السطح لأجزاء متناهية في الصغر. Català: La definició de les integrals de superfície descansa en la divisió de la superfície en petits elements de superfície. English: The definition of surface integral relies on splitting the surface into small surface elements. Figure 1: The definition of surface integral relies on splitting the surface into small surface elements. Each element is associated with a vector dS of magnitude equal to the area of the element and with direction normal to the element and pointing outward. Esperanto: La difino de surfaca integralo surbaze de disdivido de la surfaco en malgrandajn pecojn. Español: Ilustración de una superficie curvada, inmersa en , orientable y con borde; sobre la que se ha dibujado un conjunto de líneas coordenadas ortoganles. La definición de las integrales de superficie descansa en la división de la superficie en pequeños elementos de superficie. Italiano: La definizione di integrale di superficie consiste nel suddividere una superficie in parti infinitesime tanto da essere considerata piana. ქართული: სურათი 1: ზედაპირული ინტეგრალის განმარტება გულისხმობს ზედაპირის დაყოფას (უსასრულოდ) მცირე ელემენტებად. თითოეული ასეთი ელემენტი ასოცირდება dS ვექტორთან, რომლის სიდიდე ელემენტის ფართობის ტოლია, ხოლო მიმართულება ელემენტის გარე ნორმალის პარალელურია. ភាសាខ្មែរ: និយមន័យនៃអាំងតេក្រាលផ្ទៃដោយការពុះចែកផ្ទៃជាចំនែកផ្ទៃតូចៗ. Polski: Figure 1: The definition of surface integral relies on splitting the surface into small surface elements. Each element is associated with a vector dA of magnitude equal to the area of the element and with direction normal to the element and pointing outward. Shqip: Figura 1: Përcaktimi i integralit të sipërfaqes bazohet në ndarjen e sipërfaqes në elemente të vogla sipërfaqësore. Çdo element lidhet me një vektor dA me madhësi të barabarte me sipërfaqen e zonës së elementit dhe me drejtimin normal me elementin dhe i drejtuar për nga jashtë. Figure 1: Percaktimi i integralit te siperfaqes mbeshtet tek fakti qe ne mund ta ndajme siperfaqen ne elemente siperfaqesh te vogla. Cdo element lidhet me nje vektor dS me madhesi te njejte dhe te barabarte me siperfaqen e elementit dhe me drejtim perpendikular me elementin dhe i drejtuar nga jashte. Українська: Визначення поверхневого інтегралу спирається на розбиття поверхні на малі елементи. 中文: 圖一：面積分的定義需要把面分成小的面積元。每個元素跟一個向量dA聯繫，該向量的大小等於面積元的面積，而方向則是跟面積元垂直並向外。. 面积分的定义依赖于将曲面细分成小的面积元。.
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Source code ( MATLAB)

 
 
% An illustration of the surface integral.
% It shows how a surface is split into surface elements.
 
function main ()
 
% the function giving the surface and its gradient
   f=inline('10-(x.^2+y.^2)/15', 'x', 'y');
 
   BoxSize=5; % surface dimensions are 2*BoxSize x 2*BoxSize
   M = 10; % M x M = the number of surface elements into which to split the surface
   N=100;  % N x N = number of points in each surface element
   spacing = 0.1; % spacing between surface elements
   H=2*BoxSize/(M-1); % size of each surface element
   gridsize=H/N;      % distance between points on a surface element 
 
   figure(1); clf; hold on; axis equal; axis off;
 
   for i=1:(M-1)
          for j=1:(M-1)
                 Lx = -BoxSize + (i-1)*H+spacing; Ux = -BoxSize + (i  )*H-spacing;
                 Ly = -BoxSize + (j-1)*H+spacing; Uy = -BoxSize + (j  )*H-spacing;
 
%        calc the surface element
                 XX=Lx:gridsize:Ux; 
                 YY=Ly:gridsize:Uy;
                 [X, Y]=meshgrid(XX, YY);
                 Z=f(X, Y);
 
%        plot the surface element
                 surf(X, Y, Z, 'FaceColor','red', 'EdgeColor','none', ...
                          'AmbientStrength', 0.3, 'SpecularStrength', 1, 'DiffuseStrength', 0.8);
 
          end
   end
 
 
   view (-18, 40);                     % viewing angle 
   camlight headlight; lighting phong; % make nice lightning 
 
%  save to file
   print('-dpng',  '-r200', 'Surface_integral_illustration.png');

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Integral

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