# 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
on 24 Jan 2023

Commented: Bjorn Gustavsson
on 24 Jan 2023

I also would like to disp the function over the region as a plot or vector field

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### Accepted Answer

Bjorn Gustavsson
on 24 Jan 2023

Edited: Bjorn Gustavsson
on 24 Jan 2023

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

##### 2 Comments

Bjorn Gustavsson
on 24 Jan 2023

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