SamplePublisher`GrassmannCalculus`
IntegralSubstitution  |
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Details
Examples
(2)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
The following define various coordinate spaces used in the examples.
In[2]:=
xSpace:=SetEuclideanNSpace[1,{x},"Form"];uSpace:=SetEuclideanNSpace[1,{u},"Form"];Space:=SetEuclideanNSpace[1,{},"Form"]θSpace:=SetEuclideanNSpace[1,{θ},"Form"];
The following is an example of change of variable in a definite Integrate statement. The second line below is the generic substitution of variable expression.
In[3]:=
integral=Inactive[Integrate][
[]step1=template/.a0,b3,uFunction[x,1+x],fFunction[x,
x+1
,{x,0,3}]template= | IntegralSubstitution |
x
]MapAt[(#//Activate//Simplify)&,step1,2]Out[3]=
3
∫
0
1+x
xOut[3]=
b
∫
a
′
u
u[b]
∫
u[a]
Out[3]=
3
∫
0
1+x
x4
∫
1
Out[3]=
3
∫
0
1+x
x14
3
This is the same integrand done as an indefinite integral. We save the Function for back substitution to the original variable.
u
In[4]:=
integral=Inactive[Integrate][
["Indefinite"]step1=template/.u(uFunc=Function[x,1+x]),fFunction[x,
x+1
,x]template= | IntegralSubstitution |
x
]step2=MapAt[Activate,step1,2]step2/.uFunc[x]Out[4]=
∫
1+x
xOut[4]=
∫f[u[x]][x]x∫f[]
′
u
Out[4]=
∫
1+x
x∫
Out[4]=
∫
1+x
x2
3/2
3
Out[4]=
∫
1+x
x2
3
3/2
(1+x)
The following uses instead of as the variable in the initial integral.
θ
x
In[5]:=
integral=Inactive[Integrate][Tan[θ],{θ,0,π/4}]template=
[θ]step1=template/.{a0,bπ/4,uFunction[x,Tan[x]],fFunction[x,x]}MapAt[Activate,step1,2]
2
Sec[θ]
| IntegralSubstitution |
Out[5]=
π
4
∫
0
2
Sec[θ]
Out[5]=
b
∫
a
′
u
u[b]
∫
u[a]
Out[5]=
π
4
∫
0
2
Sec[θ]
1
∫
0
Out[5]=
π
4
∫
0
2
Sec[θ]
1
2
The following performs variable substitution on a .
FormIntegral
In[6]:=
integral=
x,0,+1dxtemplate=
["FormIntegral"]step1=template/.a0,b)],fFunctionu,2
,step1,2
| FormIntegral |
3
,4x2
x
| IntegralSubstitution |
3
,uFunction[x,(1+2
x
u
MapAtSpace; | EvaluateFormIntegrals |
Out[6]=
∫
ℴ
4dxx
1+
2
x
Out[6]=
∫
ℴ
′
u
∫
ℴ
Out[6]=
∫
ℴ
4dxx
1+
2
x
∫
ℴ
2d
Out[6]=
∫
ℴ
4dxx
1+
2
x
An indefinite with a variable.
FormIntegral
θ
In[7]:=
θSpace;integral=
["Indefinite",(1-Cos[3θ])Sin[3θ]dθ]template=
["Indefinite","FormIntegral",θ]step1=template/.{a0,bπ/6,u(uFunc=Function[θ,1-Cos[3θ]]),fFunction[u,u/3]}step2=MapAtSpace;
[#]+C&,step1,2step2/.uFunc[θ]
| FormIntegral |
| IntegralSubstitution |
 | EvaluateFormIntegrals |
Out[7]=
∫
ℴ
Out[7]=
∫
ℴ
′
u
∫
ℴ
Out[7]=
∫
ℴ
∫
ℴ
d
3
Out[7]=
∫
ℴ
2
6
Out[7]=
∫
ℴ
1
6
2
(1-Cos[3θ])
RowBox[{"Use", " ", "of", " ", "PullbackForms"}]
(1)
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