Vector Fields and Path Integrals


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Vector Fields:

In[]:=
F1[x_,y_]={1+2x-y^2,-2x*y};​​F2[x_,y_]={2x-y,-2x*y};​​F3[x_,y_]={0.25+y,0.5-x};​​F4[x_,y_]={-Cos[x-y]-Sin[x-y^2],Cos[x-y]+2y*Sin[x-y^2]};​​F5[x_,y_]={0.5*y/(x^2+y^2),-0.5*x/(x^2+y^2)};​​F6[x_,y_]={x,y};
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​The following command can be used to graph the vector field.
In[]:=
field1=FieldPlotter[F1[x,y],{x,y}]
Out[]=
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​When the vector field like F5 is not defined at some point, use the following, or you will get an error:
In[]:=
field5=FieldPlotter[F5[x,y],{x,y},ValueAtOrigin{0,0}]

Parametric Curves, Part I:

In[]:=
r1[t_]={t,1-t^2};​​r2[t_]={t^2-1,t^2*(2-t^2)};​​r3[t_]={-Cos[t],Sin[t]^2};
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​The following command can be used to graph the curve. Here the range of t is between -1 and 1.
In[]:=
ParametricPlot[r1[t],{t,-1,1},ColorFunctionFunction[{x,y,t},Hue[t]]]
Out[]=

Computing path integrals:

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​The following command computes the path integral of F1 over the curve described by r1 with t between -1 and 1.
In[]:=
NLineIntegrate[F6[x,y],{x,y},s4[t],{t,-1,1}]
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in t near {t} = {-0.99999996636773255628945813955289669688886178278153238352388143539}. NIntegrate obtained 1.59855×
-15
10
and 2.35206×
-10
10
for the integral and error estimates.
Out[]=
0

Parametric Curves, Part II:

In[]:=
s1[t_]={t,1-t^2};​​s2[t_]={t,.7(t-1)(t+1)(t-.7)};​​s3[t_]={(t^2)+t-1,Sin[2*Pi*t]};​​s4[t_]={t+Sin[2*Pi*t]^2,1-Exp[1-t^2]};​​
In[]:=
ParametricPlot[s4[t],{t,-1,1},ColorFunctionFunction[{x,y,t},Hue[t]]]
Out[]=

Parametric Curves, Part III:

s5[t_]={Cos[t]+(1/2)Cos[2*t],(1/2)(Sin[t]-1)+(1/6)Cos[3t]};​​s6[t_]={.8+Cos[t],0.5*Sin[t]-0.25};​​s7[t_]={Cos[2t]*Sin[t/2],1+0.5*Sin[2t]*Sin[t/2]};​​s8[t_]={t(t-2*Pi)/10,1+t(t-2*Pi)(t-1)/30};

The curl of a vector field:

Curl[F1[x,y],{x,y}]