SimReal: Mathematics - Vector Calculus - Green's Theorem - A visualization of the proof | ![]() |
A visualization of the proof of Green's theorem: By default you see the vector field F = [-y,x]. You also see a big square with corners (-2,2), (-2,-2), (2,-2) and (2,2) in the simulation window. A point (sphere) is shown in the upper left corner (-2,2) of this square together with the an arrow that shows the vector field at this position. Tangential form of Green's theorem: Mark the checkbox 'Formula Curl'. You will now see the formula of the tangential form of Green's therorem. The integral you are going to compute is the left side of this equation (shown with green color). The curve C is the contour of the big square. The integrand is the scalar product of F and the unit tangent vector T. This is equal to the tangential component Ft of F (you can see it marking the checkbox 'Ft'). This component can be positive or negative dependent on whether it points in the same direction we are going (counter-clockwise) or not. Computing the integral (green color) means that you are going to sum elements Ft*ds around the path C. Use the big scrollbar in the simulation window to travel along the curve C. Afterwards mark the checbox 'CurlDiv'. You are now going to compute the right side of the equation of Green's tangential theorem (green color). Again use the big scrollbar. Now you are travelling along all the 16 small squares that divide the big square. When travelling around one small square you are computing (in the limit of smaller and smaller squares) the k-component of the curl of F multiplied by the are of this small square. Summing all this for every small square is the same as computing the double integral over the big square (area) R (the right side of the tangential form of Green's theorem. This double integral is equal to the left side of the tangential form of Green's theorem because when travelling along all the small squares we have cancellation of every integral part except when we are travelling along the contour of the big square. All the inner curves of the small squares cancel because travelling along one such inner path is compensated travelling in the opposite direction in a neighboring small square. Normal form of Green's theorem: The same as the tangential form, except you now mark the checkbox 'Formula Div' and study the normal component Ft of F. |