SamplePublisher`GrassmannCalculus`
DirectionalDerivative  |
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Details and Options
Examples
(2)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
Work in the GrassmannPlane.
In[2]:=
 | SetActiveSpacePreferences |
 | PublicGrassmannAtlas |
| ★★V |
| ★★S |
The following are two methods for entering directional derivative operators.
In[3]:=
step1=
[,★+],[,★+]
 | DirectionalDerivative |
e
x
e
x
∇
e
x
e
x
Out[3]=
[,★+],[,★+]
∇
e
x
e
x
∇
e
x
e
x
The following evaluates on a scalar function.
In[4]:=
step1[x+y]%//Through
Out[4]=
[,★+],[,★+][x+y]
∇
e
x
e
x
∇
e
x
e
x
Out[4]=
{1,1}
A directional derivative will remain unevaluated on non-Grassmann expressions, and will evaluate when a scalar expression is substituted.
In[5]:=
∇
e
x
e
y
e
x
e
y
Out[5]=
∇
e
x
e
y
e
x
e
y
Out[5]=
3+3Cos[1]+5Sin[1]
A gives the same result after substituting the position.
VectorOperator
In[6]:=
∂[3+5][x+ySin[x]]%/.Thread
[★++]//Expand
e
x
e
y
| GrassmannCoordinates |
 | ToListCoordinates |
e
x
e
y
Out[6]=
3(1+yCos[x])+5Sin[x]
Out[6]=
3+3Cos[1]+5Sin[1]
In the case of a function that is non-differentiable at a point the two derivatives can give different results.
In[7]:=
f[x_,y_]=Piecewise{{0,x0&&y0}},y(+)
2
x
2
x
2
y
Out[7]=
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Along a diagonal from the origin the gives zero.
VectorOperator
In[8]:=
∂[+][f[x,y]]//Simplify%/.{x0,y0}
e
x
e
y
Out[8]=
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Out[8]=
0
The being a more primitive construction gives a different answer.
DirectionalDerivative
In[9]:=
∇
e
x
e
y
Out[9]=
1
2
Properties & Relations
(1)
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