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Calculate a line integral in a close path for a list of data?

Posted 6 months ago
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Please I need help doing line integral for the vector field v (z,x) components that are list of data vz and vx in specific small region (z, 0.26, 0.029), (x, 0.076, 0.079) The move direction counterclockwise.I need to do this integration in a close path. I tried to do that in a square path, so it has 4 sides. The side (a) from this point( 0.26,0.076) to ( 0.029,0.076) and the side (b) from this point ( 0.029,0.076) to ( 0.029,0.079),the side (c) ( 0.29,0.079) to ( 0.026,0.079), and (d) from( 0.026,0.079) to ( 0.026,0.076).

z = {0.025`, 0.025500000000000002`, 0.026000000000000002`, 
   0.026500000000000003`, 0.027000000000000003`, 0.0275`, 
   0.028000000000000004`, 0.0285`, 0.028999999999999998`, 
   0.029500000000000002`, 0.03`, 0.025`, 0.025500000000000002`, 
   0.026000000000000002`, 0.026500000000000003`, 
   0.027000000000000003`, 0.0275`, 0.028000000000000004`, 0.0285`, 
   0.028999999999999998`, 0.029500000000000002`, 0.03`, 0.025`, 
   0.025500000000000002`, 0.026000000000000002`, 
   0.026500000000000003`, 0.027000000000000003`, 0.0275`, 
   0.028000000000000004`, 0.0285`, 0.028999999999999998`, 
   0.029500000000000002`, 0.03`, 0.025`, 0.025500000000000002`, 
   0.026000000000000002`, 0.026500000000000003`, 
   0.027000000000000003`, 0.0275`, 0.028000000000000004`, 0.0285`, 
   0.028999999999999998`, 0.029500000000000002`, 0.03`, 0.025`, 
   0.025500000000000002`, 0.026000000000000002`, 
   0.026500000000000003`, 0.027000000000000003`, 0.0275`, 
   0.028000000000000004`, 0.0285`, 0.028999999999999998`, 
   0.029500000000000002`, 0.03`, 0.025`, 0.025500000000000002`, 
   0.026000000000000002`, 0.026500000000000003`, 
   0.027000000000000003`, 0.0275`, 0.028000000000000004`, 0.0285`, 
   0.028999999999999998`, 0.029500000000000002`, 0.03`, 0.025`, 
   0.025500000000000002`, 0.026000000000000002`, 
   0.026500000000000003`, 0.027000000000000003`, 0.0275`, 
   0.028000000000000004`, 0.0285`, 0.028999999999999998`, 
   0.029500000000000002`, 0.03`, 0.025`, 0.025500000000000002`, 
   0.026000000000000002`, 0.026500000000000003`, 
   0.027000000000000003`, 0.0275`, 0.028000000000000004`, 0.0285`, 
   0.028999999999999998`, 0.029500000000000002`, 0.03`, 0.025`, 
   0.025500000000000002`, 0.026000000000000002`, 
   0.026500000000000003`, 0.027000000000000003`, 0.0275`, 
   0.028000000000000004`, 0.0285`, 0.028999999999999998`, 
   0.029500000000000002`, 0.03`, 0.025`, 0.025500000000000002`, 
   0.026000000000000002`, 0.026500000000000003`, 
   0.027000000000000003`, 0.0275`, 0.028000000000000004`, 0.0285`, 
   0.028999999999999998`, 0.029500000000000002`, 0.03`, 0.025`, 
   0.025500000000000002`, 0.026000000000000002`, 
   0.026500000000000003`, 0.027000000000000003`, 0.0275`, 
   0.028000000000000004`, 0.0285`, 0.028999999999999998`, 
   0.029500000000000002`, 0.03`};

x = {0.075`, 0.075`, 0.075`, 0.075`, 0.075`, 0.075`, 0.075`, 0.075`, 
  0.075`, 0.075`, 0.075`, 0.0755`, 0.0755`, 0.0755`, 0.0755`, 0.0755`,
   0.0755`, 0.0755`, 0.0755`, 0.0755`, 0.0755`, 0.0755`, 0.076`, 
  0.076`, 0.076`, 0.076`, 0.076`, 0.076`, 0.076`, 0.076`, 0.076`, 
  0.076`, 0.076`, 0.0765`, 0.0765`, 0.0765`, 0.0765`, 0.0765`, 
  0.0765`, 0.0765`, 0.0765`, 0.0765`, 0.0765`, 0.0765`, 0.077`, 
  0.077`, 0.077`, 0.077`, 0.077`, 0.077`, 0.077`, 0.077`, 0.077`, 
  0.077`, 0.077`, 0.0775`, 0.0775`, 0.0775`, 0.0775`, 0.0775`, 
  0.0775`, 0.0775`, 0.0775`, 0.0775`, 0.0775`, 0.0775`, 0.078`, 
  0.078`, 0.078`, 0.078`, 0.078`, 0.078`, 0.078`, 0.078`, 0.078`, 
  0.078`, 0.078`, 0.0785`, 0.0785`, 0.0785`, 0.0785`, 0.0785`, 
  0.0785`, 0.0785`, 0.0785`, 0.0785`, 0.0785`, 0.0785`, 0.079`, 
  0.079`, 0.079`, 0.079`, 0.079`, 0.079`, 0.079`, 0.079`, 0.079`, 
  0.079`, 0.079`, 0.0795`, 0.0795`, 0.0795`, 0.0795`, 0.0795`, 
  0.0795`, 0.0795`, 0.0795`, 0.0795`, 0.0795`, 0.0795`, 0.08`, 0.08`, 
  0.08`, 0.08`, 0.08`, 0.08`, 0.08`, 0.08`, 0.08`, 0.08`, 0.08`};

for sides a and c the integration over vz with the below list:

 vz={77.05900060522465`, 135.48059457479437`, 295.6383722929741`, 
983.4858599351398`, 3279.366990725374`, 797.9335820199495`, 
259.4877468688586`, 123.68302415658533`, 71.8688421747467`, 
46.95456824451762`, 33.13259150649985`, 62.65467604965143`, 
109.76680685716522`, 239.3878557786202`, 824.2295002851661`, 
4018.612046150325`, 772.7570574840665`, 230.48596788690622`, 
106.91295953900378`, 61.37945454153419`, 39.8501684093863`, 
28.021702914760695`, 48.596990266478485`, 84.68823351012537`, 
183.8497784032177`, 644.1619587921939`, 5072.005458248049`, 
719.0637233734553`, 195.36503550079465`, 88.20619864783382`, 
50.060172522400094`, 32.31735615812771`, 22.660813028622286`, 
34.93469876754902`, 60.41482559085938`, 129.98450142732437`, 
455.20496618148894`, 6423.8053836544`, 623.4153219264374`, 
153.39968911348055`, 67.45894135046869`, 37.88499856921787`, 
24.346895425379657`, 17.045754641630154`, 21.72643849651584`, 
37.12845553069087`, 78.72439183562102`, 270.58397276530343`, 
6909.58114278669`, 467.091236333659`, 103.7819790856019`, 
44.54185099746499`, 24.814764027961303`, 15.921998469237863`, 
11.16769836607034`, 9.037929761758804`, 15.013722430477031`, 
30.91113396729486`, 101.9702210791873`, 3262.3326888176975`, 
226.0964220984996`, 45.619500322380986`, 19.29247267306306`, 
10.79387236940127`, 7.016832413064151`, 
5.012390332103408`, -3.061187202861234`, -5.751858887854979`, \
-12.758851693072359`, -42.62729458678348`, -999.7038858596454`, \
-124.83804934267619`, -22.052734264127707`, -8.492856917938184`, \
-4.253272330425062`, -2.405120063349443`, -1.4406838441080367`, \
-14.500735394328956`, -25.007446687175275`, -51.77322526412601`, \
-159.53706516454488`, -2252.873755455069`, -599.1931683098027`, \
-100.23411492030448`, -39.067236137654874`, -20.42620748003378`, \
-12.392897148490789`, -8.219239971855629`, -25.213422624998103`, \
-42.61739999598139`, -85.82243267449002`, -248.82307288892082`, \
-2281.6746771158196`, -1173.411726807893`, -189.87416734495895`, \
-72.74089310021529`, -37.85278418600245`, -23.01015564129183`, \
-15.359279988190124`, -35.138101646087236`, -58.47802262259795`, \
-114.79792602285382`, -313.04787663472194`, -2042.725772575267`, \
-1762.9409712210654`, -291.7137023328667`, -109.88715481882755`, \
-56.69401378458309`, -34.33760652370107`, -22.906368035853674`, \
-44.2226715475026`, -72.52201721791879`, -138.77302249905966`, \
-356.01623781182496`, -1780.340821113502`, -2238.4644221762014`, \
-406.0153112876721`, -150.9445993066379`, -77.14938212269567`, \
-46.47587498120036`, -30.917155216906792`};

and for side b and d the integration over vx:

 vx = {-55.67007990547937`, -75.14597114885453`, -113.57719742504118`, \
  -208.36341120868562`, -126.64314947803405`, 107.4544792189494`, 
   79.69580331345534`, 59.11407106921598`, 46.453865515004296`, 
   38.11339159427955`, 
   32.24990222186154`, -54.96258268613943`, -74.0036655994076`, \
   -111.87479935046294`, -212.57084955700506`, -189.4336160159972`, 
   126.50777170380746`, 86.07242066587635`, 62.1068605625609`, 
  48.18758390926222`, 39.252407294901225`, 
  33.061847489248436`, -54.20994350884543`, -72.72924561369581`, \
 -109.61494189786147`, -212.35296496166256`, -309.4429590888888`, 
  149.43681411788697`, 92.79755790085518`, 65.16520449320554`, 
   49.94021063654884`, 40.39882272480382`, 
  33.87756734943816`, -53.38320981813595`, -71.27996108245262`, \
 -106.74182560124207`, -207.2822338269145`, -550.6567423445471`, 
  177.27032354781733`, 99.95506805688883`, 68.3337146954808`, 
  51.740318750610456`, 41.57223751827236`, 
  34.71121531151899`, -52.45686335328409`, -69.62131892736366`, \
  -103.23287770655737`, -197.59682310263753`, -966.2245119620898`, 
   211.08896975633735`, 107.64743849230592`, 71.66665456205098`, 
   53.62123222631994`, 42.794854285274546`, 
   35.578510996648745`, -51.40938475924947`, -67.72833356797355`, \
  -99.10034248662822`, -184.12688963365042`, -1145.4552426634589`, 
   251.63994734081342`, 115.99552098121745`, 75.22880313037936`, 
   55.62143134820425`, 44.09168987103588`, 
   36.49686357794036`, -50.22377290086338`, -65.58637717248659`, \
  -94.38937226407029`, -168.05103871973063`, -750.3780570837548`, 
   298.4402447186256`, 125.13580243113759`, 79.0964322900108`, 
   57.78510851896807`, 45.49088561303157`, 
   37.4855585164117`, -48.887965265983404`, -63.1915182613739`, \
  -89.17299142643907`, -150.60613124878205`, -405.09590087812785`, 
   348.2009259152962`, 135.2134752219103`, 83.35838196866278`, 
   60.16291188478557`, 47.024140161267134`, 
   38.5660211492725`, -47.39511158689412`, -60.550290590742534`, \
  -83.5448332815952`, -132.8650709110221`, -225.57015008140263`, 
    393.1893276961692`, 146.36866783354998`, 88.11715423942269`, 
    62.81291255349221`, 48.72729114089431`, 
   39.76217281411414`, -45.74366415053528`, -57.678898459937855`, \
  -77.61083470156927`, -115.62903449707156`, -132.18461472301797`, 
   421.9132226897307`, 158.7121324392396`, 93.48983224950746`, 
    65.80182524037085`, 50.641075187404454`, 
    41.100898102966575`, -43.93726410404289`, -54.6019280586431`, \
   -71.48106889937198`, -99.41759267055859`, -79.33261726845735`, 
    424.7090681538802`, 172.28586637279076`, 99.6084328049799`, 
    69.20649799594959`, 52.81209937187389`, 42.61264565885176`};

I tried to do this integration:

In[722]:= fz = Interpolation[vz]

a = NIntegrate[fz[z], {z, 0.026`, 0.029}]

Out[722]= InterpolatingFunction[{{1., 121.}}, <>]

Out[723]= -0.860932
In[724]:= c = NIntegrate[fz[z], {z, 0.029, 0.026}]

Out[724]= 0.860932
In[725]:= fx = Interpolation[vx]

b = NIntegrate[fx[x], {x, 0.076, 0.079}]

Out[725]= InterpolatingFunction[{{1., 121.}}, <>]

Out[726]= -0.0666142
In[727]:= d = NIntegrate[fx[x], {x, 0.079, 0.076}]

Out[727]= 0.0666142

In[728]:= a + b + c + d

Out[728]= 0.

I see that the interpolation out of the small grid region, and the result of a + b + c + d should be 6.28319 which equals 2 pi, but the result is 0. Do you have any suggestions to get the result (2 pi).

Thanks in advance

2 Replies

What is your vector-field? It should be a two-component-function vz[ z, x ] := { c1[ z, x ], c2 [ z, x ] } If this is constant you of course get 0 as result of your line-integral. Note in your calculation above a = -c and b = - d.

vz must be different on the different parts of the integration-path

Posted 6 months ago

Thank you so much !

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