# Integrating a line integral e^x(sinydx + cosydy) over an ellipse 4(x+1)^2 + 9(y-3)^2 = 36

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Yuva il 24 Gen 2023
Commentato: Bjorn Gustavsson il 24 Gen 2023
I also would like to disp the function over the region as a plot or vector field
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Bjorn Gustavsson il 24 Gen 2023
Modificato: Bjorn Gustavsson il 24 Gen 2023
For the integration you should use Green's theorem. It is beautiful, especially for this case.
For the vector-field-plot you can use quiver, see the help and documentation for that function. There are also a couple of color-enhanced variations available on the file exchange: quiver-magnitude-dependent-color-in-2d-and-3d, cquiver, ncquiverref and quiverc (it is rather likely that I've missed some variant, but you can search on further). You could do something like:
phi360 = linspace(0,2*pi,361);
x0 = -1;
y0 = 3;
xE = x0 + sqrt(36/4)*cos(phi360);
yE = y0 + sqrt(36/9)*sin(phi360);
plot(xE,yE,'k','linewidth',2)
[x,y] = meshgrid(-4.5:0.1:2.5,0.5:0.1:5.5);
fx = @(x,y) exp(x).*sin(y);
fy = @(x,y) exp(x).*cos(y);
quiver(x,y,fx(x,y),fy(x,y)) % Either of these 4 calls to quiver, or with some
quiver(x,y,fx(x,y),fy(x,y),1) % normalization of your own, I like the color-
quiver(x,y,fx(x,y),fy(x,y),0) % capable extensions, because then one can
quiver(x(1:5:end,1:5:end),... % plot the unit-vectors of the direction of
y(1:5:end,1:5:end),... % the forces and have their magnitude in color
fx(x(1:5:end,1:5:end),y(1:5:end,1:5:end)),...
fy(x(1:5:end,1:5:end),y(1:5:end,1:5:end)),0)
for i1 = 1:10:numel(phi360)
xC = xE(i1);
yC = yE(i1)
FxC = fx(xC,yC);
FyC = fy(xC,yC);
arrow3([xC,yC],[xC,yC]+[FxC,FyC]) % or arrow, both available on the FEX
end
You now have a solution to your task. If you look up the Green's theorem link on Wikipedia you should also make an additional pseudocolor-plot, likely put that one first in the script. You should also comment and work out exactly what happens on each line. (the normalization of quiver is a bit fiddly to get a nice and ballanced figure)
HTH
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Yuva il 24 Gen 2023
i added a hold on before the for to still have the previous plot
Bjorn Gustavsson il 24 Gen 2023
@Yuva, good that it helped. The answer was a bit quick. When it comes to graphics it is possible to further decorate and combine different presentations to make better figures.

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