Numerical calculation of line integral over a vector field

Question

Amir Kleiner 2019-1-25

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https://ww2.mathworks.cn/matlabcentral/answers/441416-numerical-calculation-of-line-integral-over-a-vector-field

编辑： Torsten 2019-1-28

在 MATLAB Online 中打开

Hey,

I have a path given by three vectors

and a vector field also given by three vectors, evaluated only along the path -

.

I need to find the line integral

.

My problem is that the length of

is different from the length of

and if I truncate it in different form I get very different results:

% The vector field is Fx, Fy, Fz
% The path is rx, ry, rz
% the time vector is t
% all 7 vectors have the same size.
dt = diff(t);
drx=diff(rx)./dt; dry=diff(ry)./dt; drz=diff(rz)./dt;
%% 1st try:
Fx=Fx(1:end-1); Fy=Fy(1:end-1); Fz=Fz(1:end-1);
t = t(1:end-1);
%%2nd try:
Fx=Fx(2:end); Fy=Fy(2:end); Fz=Fz(2:end);
t=t(2:end);
%% integral calculation
Fdr = Fx.*drx + Fy.*dry + Fz.*drz;
W = cumtrapz(t,Fdr)
plot(t,W)

The 1st try code results in an increasing function, while the 2nd try code results in a decreasing function.

I'd appreciate some help with this integration.

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Answer 1

Torsten 2019-1-25

0
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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/441416-numerical-calculation-of-line-integral-over-a-vector-field#answer_357958

The usual way is to approximate the integral as

sum_{i=1}^{n-1} [ (rx(t(i+1))-rx(t(i))/(t(i+1)-t(i)) * Fx(gamma((t(i+1)+t(i))/2) +  ...
                  (ry(t(i+1))-ry(t(i))/(t(i+1)-t(i)) * Fy(gamma((t(i+1)+t(i))/2) + ...
                  (rz(t(i+1))-rz(t(i))/(t(i+1)-t(i)) * Fz(gamma((t(i+1)+t(i))/2)  ]

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Amir Kleiner 2019-1-26

在 MATLAB Online 中打开

I don't really understand your unswer...

By:

Fx(gamma((t(i+1)+t(i))/2)

Do you mean:

(Fx(gamma(t(i+1)))+Fx(gamma(t(i))))/2

Or do you mean something else?

Since both the path and the vector field are given only at specific time, I can't really use the value at an intermediate time.

Also, this seems like a Riemann approximation for the integral, which is fine, but doesn't really unswer the question of how to calculate this integral in MatLab...

Torsten 2019-1-28

编辑：Torsten 2019-1-28

>By:

>Fx(gamma((t(i+1)+t(i))/2)

>Do you mean:

>(Fx(gamma(t(i+1)))+Fx(gamma(t(i))))/2

Yes, that's what I mean.

> Also, this seems like a Riemann approximation for the integral, which is fine, but doesn't really unswer the question of how to calculate this integral in MatLab...

Sums can also programmed in MATLAB ...

But this is shorter:

dt = diff(t);
drx=diff(rx)./dt; 
dry=diff(ry)./dt; 
drz=diff(rz)./dt;
t = (t(1:end-1)+t(2:end))/2.0;
Fx = (Fx(1:end-1)+Fx(2:end))/2.0;
Fy = (Fy(1:end-1)+Fy(2:end))/2.0;
Fz = (Fz(1:end-1)+Fz(2:end))/2.0;
%% integral calculation
Fdr = Fx.*drx + Fy.*dry + Fz.*drz;
W = cumtrapz(t,Fdr)
plot(t,W)