SamplePublisher`GrassmannCalculus`
EvaluateFormIntegrals  |
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Details and Options
Examples
(15)
Basic Examples
(1)
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<<GrassmannCalculus`
To find the area between two curves;
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 | SetCoordinateVectorSpace |
| FormIntegral |
 | EvaluateFormIntegrals |
Out[2]=
∫
ℴ
Out[2]=
2
2
The following form of the integral, using domain iterators, is more easily evaluated. Show the unevaluated Mathematica Integrate statement.
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domain={{x,π/4,5π/4},{y,Cos[x],Sin[x]}};
[domain,dx⋀dy]
[%,InactiveIntegralTrue]Activate[%]
| FormIntegral |
| EvaluateFormIntegrals |
Out[3]=
∫
ℴ
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5π
4
∫
π
4
Sin[x]
∫
Cos[x]
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2
2
Evaluate an integral that uses a Region and a parameter. Without an Assumptions option we obtain a . We also show the unevaluated statement generated.
ConditionalExpression
Integrate
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| FormIntegral |
| EvaluateFormIntegrals |
Out[4]=
∫
ℴ
Out[4]=
ConditionalExpression[π,R>0]
2
R
In the following we add an Assumption.
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| FormIntegral |
| EvaluateFormIntegrals |
Out[5]=
∫
ℴ
Out[5]=
π
2
R
The following uses a .
ParametricRegion
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| SetCoordinateVectorSpace |
2
x
| FormIntegral |
| EvaluateFormIntegrals |
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ParametricRegion[{{x,-y+(1+y)},-1≤x≤1&&0≤y≤1},{x,y}]
2
x
Out[6]=
∫
ℴ
Out[6]=
4
3
The following integrates a Gaussian over a full 1-dimensional space.
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| SetCoordinateVectorSpace |
| FormIntegral |
2
x
| EvaluateFormIntegrals |
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∫
ℴ
-
2
x
2
Out[7]=
2π
The following integrates over a discretized disk.
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| SetCoordinateVectorSpace |
| FormIntegral |
| EvaluateFormIntegrals |
Out[8]=
∫
ℴ
Out[8]=
Out[8]=
0.785398
In this case, it did not have to be discretized.
In[9]:=
| SetCoordinateVectorSpace |
| FormIntegral |
| EvaluateFormIntegrals |
Out[9]=
∫
ℴ
Out[9]=
π
4
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| SetCoordinateVectorSpace |
| FormIntegral |
| EvaluateFormIntegrals |
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ParametricRegion[{{x,Sin[x]},0≤x≤π},{x}]
Out[10]=
∫
ℴ
Out[10]=
2
2
EllipticE1
2
Here is a line integral using a . We need a 2-dimensional space even though it's a 1-dimensional integral.
ParametricRegion
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| SetCoordinateVectorSpace |
| FormIntegral |
| EvaluateFormIntegrals |
Out[11]=
∫
ℴ
Out[11]=
2
2
EllipticE1
2
Out[11]=
3.8202
The following is a Gaussian integrated over a square centered at the origin.
In[12]:=
| SetCoordinateVectorSpace |
| FormIntegral |
2
x
2
y
| EvaluateFormIntegrals |