In this chapter some examples are provided.
4.1 Calculus in \(\mathbb{R}^3\)
We will prove the following identities (see the summary, 4.1.4, for details):
\[\begin{split}d\,f = [\mathtt{grad}\,f]_1 \\
d\,[T]_1 = [\mathtt{curl}\,T]_2 \\
d\,[T]_2 = [\mathtt{div}\,T]_3\end{split}\]
Let M denote our differential graded algebra on \(\mathbb{R}^3\). In
FriCAS we can express this as
\[\mathtt{DifferentialForms(Integer,[x,y,z])}\]
Type: Type
The list of available methods can be seen by
)show M
DifferentialForms(Integer,[x,y,z]) is a package constructor.
Abbreviation for DifferentialForms is DFORM
This constructor is exposed in this frame.
------------------------------- Operations --------------------------------
coordSymbols : () -> List(Symbol)
...
baseForms : () -> List(DeRhamComplex(Integer,[x,y,z]))
coordVector : () -> List(Expression(Integer))
.
.
.
The position vector \(P=(x,y,z)\) and the basis of one forms can be
obtained by
\[[x,y,z]\]
Type: List(Expression(Integer))
and
\[[dx,dy,dz]\]
Type: List(DeRhamComplex(Integer,[x,y,z]))
This way we can call the coordinates as \(P.i\) and the basis one forms
as \(dP.i\). Of course, we can also use \(dx,dy,dz\) directly when
we set
[dx,dy,dz]:=baseForms()$M
or when we use the generators of the domain DeRhamComplex itself:
dx:=generator(1)$DERHAM(INT,[x,y,z])
dy:= ...
The first method, however, is quite convenient when using indexed coordinates
and also because we can form expressions like
\[z\ dz + y\ dy + x\ dx\]
Type: DeRhamComplex(Integer,[x,y,z]).
4.1.1 Gradient
There are many ways to create a zero form, one of those is
\[f(x,y,z)\]
Type: DeRhamComplex(Integer,[x,y,z])
Now we apply the exterior differential operator to f:
\[{{{f _ {{,3}}}
\left(
{x, \: y, \: z}
\right)}
\ dz}+{{{f _ {{,2}}}
\left(
{x, \: y, \: z}
\right)}
\ dy}+{{{f _ {{,1}}}
\left(
{x, \: y, \: z}
\right)}
\ dx}\]
Type: DeRhamComplex(Integer,[x,y,z])
The coefficients of \(df\) are just
[coefficient(d f, dP.j) for j in 1..3]
\[ \left[
{{f _ {{,1}}}
\left(
{x, \: y, \: z}
\right)},
\: {{f _ {{,2}}}
\left(
{x, \: y, \: z}
\right)},
\: {{f _ {{,3}}}
\left(
{x, \: y, \: z}
\right)}
\right]\]
Type: List(Expression(Integer))
the components of the gradient vector \(\nabla f\) of \(f\).
4.1.2 Curl
Let T be a generic vector field on \(M=\mathbb{R}^3\):
\[\left[
{{T _ {1}}
\left(
{x, \: y, \: z}
\right)},
\: {{T _ {2}}
\left(
{x, \: y, \: z}
\right)},
\: {{T _ {3}}
\left(
{x, \: y, \: z}
\right)}
\right]\]
Type: List(Expression(Integer))
Then we build the general one form \(\tau\):
Now we apply the exterior differential operator \(d\):
\[\begin{split}\scriptstyle{
{{\left( {{{T _ {3}} _ {{,2}}}
\left(
{x, \: y, \: z}
\right)}
-{{{T _ {2}} _ {{,3}}}
\left(
{x, \: y, \: z}
\right)}
\right)}
\ dy \ dz}+{{\left( {{{T _ {3}} _ {{,1}}}
\left(
{x, \: y, \: z}
\right)}
-{{{T _ {1}} _ {{,3}}}
\left(
{x, \: y, \: z}
\right)}
\right)}
\ dx \ dz}+ \\ {{\left( {{{T _ {2}} _ {{,1}}}
\left(
{x, \: y, \: z}
\right)}
-{{{T _ {1}} _ {{,2}}}
\left(
{x, \: y, \: z}
\right)}
\right)}
\ dx \ dy}
}\end{split}\]
Type: DeRhamComplex(Integer,[x,y,z])
Next, we want to extract the coefficients:
[coefficient(d tau, m) for m in monomials(2)$M]
\[\scriptstyle{
\left[
{{{{T _ {2}} _ {{,1}}}
\left(
{x, \: y, \: z}
\right)}
-{{{T _ {1}} _ {{,2}}}
\left(
{x, \: y, \: z}
\right)}},
\: {{{{T _ {3}} _ {{,1}}}
\left(
{x, \: y, \: z}
\right)}
-{{{T _ {1}} _ {{,3}}}
\left(
{x, \: y, \: z}
\right)}},
\: {{{{T _ {3}} _ {{,2}}}
\left(
{x, \: y, \: z}
\right)}
-{{{T _ {2}} _ {{,3}}}
\left(
{x, \: y, \: z}
\right)}}
\right]}\]
The (well known) curl is defined as
\[\mathtt{curl}(T) =\nabla\times T= \scriptstyle{
\left(
\frac{\partial T_3}{\partial y} - \frac{\partial T_2}{\partial z},
\frac{\partial T_1}{\partial z} - \frac{\partial T_3}{\partial x},
\frac{\partial T_2}{\partial x} - \frac{\partial T_1}{\partial y}
\right)}\]
curl(V) == [D(V.3,y)-D(V.2,z),D(V.1,z)-D(V.3,x),D(V.2,x)-D(V.1,y)]
We now claim that the following identity holds:
\[d (T\, dP) = \star(\mathtt{curl}(V)\, dP)\]
where * denotes the Hodge star operator with respect to the Euclidean
metric
g:=diagonalMatrix([1,1,1])
\[\begin{split}\left[
\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}
\right]\end{split}\]
To prove it we just have to test:
test( d(T*dP) = hodgeStar(g,curl(T)*dP)$M )
\[\mathtt{true}\]
Type: Boolean
4.1.3 Divergence
Again, let T be a generic vector field on \(M=\mathbb{R}^3\), then the
divergence is defined by
\[\mathtt{div}(T) = \nabla \bullet T =
\scriptstyle{
\frac{\partial T_1}{\partial x} +
\frac{\partial T_2}{\partial y} +
\frac{\partial T_3}{\partial z}}.\]
When we calculate
we get the 3-form
\[{\left( {{{T _ {3}} _ {{,3}}}
\left(
{x, \: y, \: z}
\right)}+{{{T
_ {2}} _ {{,2}}}
\left(
{x, \: y, \: z}
\right)}+{{{T
_ {1}} _ {{,1}}}
\left(
{x, \: y, \: z}
\right)}
\right)}
\ dx \ dy \ dz\]
4.1.4 Summary
Let us summarize what we have obtained above. We use the following notation
for the mapping of scalar functions and vector fields to differential forms:
\[\begin{split}f \rightarrow [f]_0 \\
T \rightarrow [T]_1\end{split}\]
where the index denotes the degree of the form. Moreover, we define another
pair of forms by applying the Hodge operator:
\[\begin{split}[T]_2 = \star [T]_1 \\
[f]_3 = \star [f]_0\end{split}\]
So we can state the general identities:
\[\begin{split}d\,f = [\nabla\,f]_1 \\
d\,[T]_1 = [\mathtt{curl}\,T]_2 \\
d\,[T]_2 = [\mathtt{div}\,T]_3\end{split}\]
4.1.5 Hodge duals
To conclude this example, we are going to calculate a table for the Hodge
duals of the monomials.
g:=diagonalMatrix([1,1,1])::SquareMatrix(3,INT)
[[hodgeStar(g,m)$M for m in monomials(j)$M] for j in 0..3]
\[\left[
{\left[ {dx \ dy \ dz}
\right]},
\: {\left[ {dy \ dz}, \: -{dx \ dz}, \: {dx \ dy}
\right]},
\: {\left[ dz, \: -dy, \: dx
\right]},
\: {\left[ 1
\right]}
\right]\]
Type: List(List(DeRhamComplex(Integer,[x,y,z])))
Thus we get the following table:
| \(\alpha\) |
\(\star\alpha\) |
\(\star\star\alpha\) |
|---|
| \(1\) |
\(dx\wedge dy \wedge dz\) |
\(1\) |
| \(dx\) |
\(dy \wedge dz\) |
\(dx\) |
| \(dy\) |
\(-dx \wedge dz\) |
\(dy\) |
| \(dz\) |
\(dx \wedge dy\) |
\(dz\) |
By the way, this method can be applied in any dimension for any metric.
4.2 Faraday 2-form
The free electromagnetic field can be described by a 2-form F in
Minkowski space. This form - also known as Faraday 2-form - is given by
\[\scriptstyle{
F=B_1\ dy\wedge dz + B_2\ dz\wedge dx + B_3\ dx\wedge dy +
E_1\ dx\wedge dt + E_2\ dy\wedge dt + E_3\ dz\wedge dt
}\]
where we here use the cgs system and E, B denote the classical
fields (see the example in the documentation of DeRhamComplex).
To represent F in FriCAS we have to choose space-time variables
\(x,y,z,t\), in the correct order, and \(g\) will be the
Minkowski metric:
v := [x,y,z,t]
g := diagonalMatrix([-1,-1,-1,1])::SquareMatrix(4,INT)
M := DFORM(INT,v)
R ==> EXPR(INT)
Instead of \(x,y,z,t\) we also could have chosen \(x_0,x_1,x_2,x_3\)
for instance. Now we need the coordinates and basis one forms:
- Important
- The order of the variables must coincide with that in the metric g.
That means for example, for \(t,x,y,z\) the positive
1 comes
first.
X := coordVector()$M
dX := baseForms()$M
We also need the field E and B, but this time we will not choose the
vectorField function because we only need three components:
E := [operator E[i] for i in 1..3]
B := [operator B[i] for i in 1..3]
Eventually we can build F:
F := (B.1 X)*dX.2*dX.3 + (B.2 X)*dX.3*dX.1 + (B.3 X)*dX.1*dX.2 +_
(E.1 X)*dX.1*dX.4 + (E.2 X)*dX.2*dX.4 + (E.3 X)*dX.3*dX.4
\[\begin{split}\scriptstyle{
{{{E _ {3}}
\left(
{x, \: y, \: z, \: t}
\right)}
\ dz \ dt}+{{{E _ {2}}
\left(
{x, \: y, \: z, \: t}
\right)}
\ dy \ dt}+{{{B _ {1}}
\left(
{x, \: y, \: z, \: t}
\right)}
\ dy \ dz} + \\
{{{E _ {1}}
\left(
{x, \: y, \: z, \: t}
\right)}
\ dx \ dt} -{{{B _ {2}}
\left(
{x, \: y, \: z, \: t}
\right)}
\ dx \ dz}+{{{B _ {3}}
\left(
{x, \: y, \: z, \: t}
\right)}
\ dx \ dy}
}\end{split}\]
Type: DeRhamComplex(Integer,[x,y,z,t])
We apply the exterior differential operator d to F:
\[\begin{split}\scriptstyle{
{{\left( {{{E _ {3}} _ {{,2}}}
\left(
{x, \: y, \: z, \: t}
\right)}
-{{{E _ {2}} _ {{,3}}}
\left(
{x, \: y, \: z, \: t}
\right)}+{{{B
_ {1}} _ {{,4}}}
\left(
{x, \: y, \: z, \: t}
\right)}
\right)}
\ dy \ dz \ dt}\, + \\
{{\left( {{{E _ {3}} _ {{,1}}}
\left(
{x, \: y, \: z, \: t}
\right)}
-{{{E _ {1}} _ {{,3}}}
\left(
{x, \: y, \: z, \: t}
\right)}
-{{{B _ {2}} _ {{,4}}}
\left(
{x, \: y, \: z, \: t}
\right)}
\right)}
\ dx \ dz \ dt}\, + \\
{{\left( {{{E _ {2}} _ {{,1}}}
\left(
{x, \: y, \: z, \: t}
\right)}
-{{{E _ {1}} _ {{,2}}}
\left(
{x, \: y, \: z, \: t}
\right)}+{{{B
_ {3}} _ {{,4}}}
\left(
{x, \: y, \: z, \: t}
\right)}
\right)}
\ dx \ dy \ dt}\, + \\
{{\left( {{{B _ {3}} _ {{,3}}}
\left(
{x, \: y, \: z, \: t}
\right)}+{{{B
_ {2}} _ {{,2}}}
\left(
{x, \: y, \: z, \: t}
\right)}+{{{B
_ {1}} _ {{,1}}}
\left(
{x, \: y, \: z, \: t}
\right)}
\right)}
\ dx \ dy \ dz}
}\end{split}\]
Type: DeRhamComplex(Integer,[x,y,z,t])
We see at once that the first three terms of the sum correspond to the
vector
\[\nabla\times\mathbf{E}+\frac{\partial\mathbf{B}}{\partial t}\]
and the fourth term is
\[\nabla\bullet\mathbf{B}.\]
Actually, all terms are zero by two of the Maxwell equations. Consequently
we have shown (the well known fact)
\[d\mathbf{F} = 0\]
Now let us apply the \(\star\)-operator to F, which is also a 2-form:
\[\begin{split}\scriptstyle{
{{{B _ {3}}
\left(
{x, \: y, \: z, \: t}
\right)}
\ dz \ dt} + {{{B _ {2}}
\left(
{x, \: y, \: z, \: t}
\right)}
\ dy \ dt} -{{{E _ {1}}
\left(
{x, \: y, \: z, \: t}
\right)}
\ dy \ dz}+ \\
{{{B _ {1}}
\left(
{x, \: y, \: z, \: t}
\right)}
\ dx \ dt} + {{{E _ {2}}
\left(
{x, \: y, \: z, \: t}
\right)}
\ dx \ dz}- {{{E _ {3}}
\left(
{x, \: y, \: z, \: t}
\right)}
\ dx \ dy}
}\end{split}\]
Type: DeRhamComplex(Integer,[x,y,z,t])
Now, as before:
\[\begin{split}\scriptstyle{
{{\left( -{{{E _ {1}} _ {{,4}}}
\left(
{x, \: y, \: z, \: t}
\right)}+{{{B
_ {3}} _ {{,2}}}
\left(
{x, \: y, \: z, \: t}
\right)}
-{{{B _ {2}} _ {{,3}}}
\left(
{x, \: y, \: z, \: t}
\right)}
\right)}
\ dy \ dz \ dt}+ \\
{{\left( {{{E _ {2}} _ {{,4}}}
\left(
{x, \: y, \: z, \: t}
\right)}+{{{B
_ {3}} _ {{,1}}}
\left(
{x, \: y, \: z, \: t}
\right)}
-{{{B _ {1}} _ {{,3}}}
\left(
{x, \: y, \: z, \: t}
\right)}
\right)}
\ dx \ dz \ dt}+ \\
{{\left( -{{{E _ {3}} _ {{,4}}}
\left(
{x, \: y, \: z, \: t}
\right)}+{{{B
_ {2}} _ {{,1}}}
\left(
{x, \: y, \: z, \: t}
\right)}
-{{{B _ {1}} _ {{,2}}}
\left(
{x, \: y, \: z, \: t}
\right)}
\right)}
\ dx \ dy \ dt}+ \\
{{\left( -{{{E _ {3}} _ {{,3}}}
\left(
{x, \: y, \: z, \: t}
\right)}
-{{{E _ {2}} _ {{,2}}}
\left(
{x, \: y, \: z, \: t}
\right)}
-{{{E _ {1}} _ {{,1}}}
\left(
{x, \: y, \: z, \: t}
\right)}
\right)}
\ dx \ dy \ dz}
}\end{split}\]
Type: DeRhamComplex(Integer,[x,y,z,t])
Again, we see that the first three terms correspond to
\[-\frac{\partial\mathbf{E}}{\partial t}+ \nabla\times\mathbf{B}\]
while the last one corresponds to:
\[-\,\nabla\bullet\mathbf{E}\]
Thus, in vacuum, these are the second pair of Maxwell’s equation and we
have:
\[d \star\mathbf{F} = 0\]
To conclude this example we will compute the quantities (4-forms):
\[\mathbf{F} \wedge \mathbf{F} \ \ \mathrm{and} \ \
\mathbf{F} \wedge \star\mathbf{F}.\]
Recalling the definition of the Hodge dual it is sufficient (in principle)
to compute the scalar product \(\langle F,F\rangle\):
\[\begin{split}\scriptstyle{
-{{{{E _ {3}}
\left(
{x, \: y, \: z, \: t}
\right)}}
^ {2}} -{{{{E _ {2}}
\left(
{x, \: y, \: z, \: t}
\right)}}
^ {2}} -{{{{E _ {1}}
\left(
{x, \: y, \: z, \: t}
\right)}}
^ {2}}+ \\
{{{{B _ {3}}
\left(
{x, \: y, \: z, \: t}
\right)}}
^ {2}}+{{{{B _ {2}}
\left(
{x, \: y, \: z, \: t}
\right)}}
^ {2}}+{{{{B _ {1}}
\left(
{x, \: y, \: z, \: t}
\right)}}
^ {2}}
}\end{split}\]
Type: Expression(Integer)
and \(\langle F,\star F\rangle\):
\[\scriptstyle{
-{2 \ {{B _ {3}}
\left(
{x, \: y, \: z, \: t}
\right)}
\ {{E _ {3}}
\left(
{x, \: y, \: z, \: t}
\right)}}
-{2 \ {{B _ {2}}
\left(
{x, \: y, \: z, \: t}
\right)}
\ {{E _ {2}}
\left(
{x, \: y, \: z, \: t}
\right)}}
-{2 \ {{B _ {1}}
\left(
{x, \: y, \: z, \: t}
\right)}
\ {{E _ {1}}
\left(
{x, \: y, \: z, \: t}
\right)}}
}\]
Type: Expression(Integer)
Indeed, we can test the defining identity, e.g. for the first case:
test(F * %F = dot(g,F,F)$M * volumeForm(g)$M)
\[\mathtt{true}\]
Type: Boolean
4.4 More examples (way of working)
)clear all
All user variables and function definitions have been cleared.
n:=4 -- dim of base space (n>=2 !)
R:=Integer -- Ring
v:=[subscript(x,[j::OutputForm]) for j in 1..n] -- (x_1,..,x_n)
M:=DFORM(R,v)
-- basis 1-forms and coordinate vector
dx:=baseForms()$M -- [dx[1],...,dx[n]]
x:=coordVector()$M -- [x[1],...,x[n]]
xs:=coordSymbols()$M -- as above but as List Symbol (for differentiate, D)
-- operator, vector field, scalar field, symbol
a:=operator 'a -- operator
b:=vectorField(b)$M -- generic vector field [b1(x1..xn),...,bn(x1..xn)]
c:=vectorField(c)$M
P:=scalarField(P)$M -- scalar field P(x1,..,xn)
-- metric
g:=diagonalMatrix([1 for i in 1..n])$SquareMatrix(n,EXPR R) -- Euclidean
h:=diagonalMatrix(c)$SquareMatrix(n,EXPR R)
-- vector field (R)
vf:=vector b
-- macros
dV(g) ==> volumeForm(g)$M
i(X,w) ==> interiorProduct(X,w)$M
L(X,w) ==> lieDerivative(X,w)$M
** w ==> hodgeStar(g,w)$M -- don't use * instead of ** !
\[{{x _ {1}} \ {dx _ {2}}} -{{x _ {2}} \ {dx _ {1}}}\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[2 \ {dx _ {1}} \ {dx _ {2}}\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[0\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[{{x _ {1}} \ {{b _ {2}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
-{{x _ {2}} \ {{b _ {1}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[\begin{split}{{\left( {{x _ {1}} \ {{{b _ {2}} _ {{,4}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
-{{x _ {2}} \ {{{b _ {1}} _ {{,4}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
\right)}
\ {dx _ {4}}}+ \\ {{\left( {{x _ {1}} \ {{{b _ {2}} _ {{,3}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
-{{x _ {2}} \ {{{b _ {1}} _ {{,3}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
\right)}
\ {dx _ {3}}}+ \\ {{\left( {{x _ {1}} \ {{{b _ {2}} _ {{,2}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
-{{x _ {2}} \ {{{b _ {1}} _ {{,2}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}+{{b
_ {1}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\right)}
\ {dx _ {2}}}+ \\ {{\left( {{x _ {1}} \ {{{b _ {2}} _ {{,1}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
-{{x _ {2}} \ {{{b _ {1}} _ {{,1}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
-{{b _ {2}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\right)}
\ {dx _ {1}}}\end{split}\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[\begin{split}{{\left( {{x _ {1}} \ {{{b _ {2}} _ {{,4}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
-{{x _ {2}} \ {{{b _ {1}} _ {{,4}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
\right)}
\ {dx _ {4}}}+ \\ {{\left( {{x _ {1}} \ {{{b _ {2}} _ {{,3}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
-{{x _ {2}} \ {{{b _ {1}} _ {{,3}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
\right)}
\ {dx _ {3}}}+ \\ {{\left( {{x _ {1}} \ {{{b _ {2}} _ {{,2}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
-{{x _ {2}} \ {{{b _ {1}} _ {{,2}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}+{{b
_ {1}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\right)}
\ {dx _ {2}}}+ \\ {{\left( {{x _ {1}} \ {{{b _ {2}} _ {{,1}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
-{{x _ {2}} \ {{{b _ {1}} _ {{,1}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
-{{b _ {2}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\right)}
\ {dx _ {1}}}\end{split}\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[0\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[{{{x _ {2}}} ^ {2}}+{{{x _ {1}}} ^ {2}}\]
Type: Expression(Integer)
d i(vf,dV(g)) -- div(b) dV
\[\def\sp{^}\def\sb{_}\def\leqno(#1){}
\def\erf{\mathrm{erf}}\def\sinh{\mathrm{sinh}}
\def\zag#1#2{{{\left.{#1}\right|}\over{\left|{#2}\right.}}}
\def\csch{\mathrm{csch}}
{\left( {{{b \sb {4}} \sb {{,4}}}
\left(
{{x \sb {1}}, \: {x \sb {2}}, \: {x \sb {3}}, \: {x \sb {4}}}
\right)}+{{{b
\sb {3}} \sb {{,3}}}
\left(
{{x \sb {1}}, \: {x \sb {2}}, \: {x \sb {3}}, \: {x \sb {4}}}
\right)}+{{{b
\sb {2}} \sb {{,2}}}
\left(
{{x \sb {1}}, \: {x \sb {2}}, \: {x \sb {3}}, \: {x \sb {4}}}
\right)}+{{{b
\sb {1}} \sb {{,1}}}
\left(
{{x \sb {1}}, \: {x \sb {2}}, \: {x \sb {3}}, \: {x \sb {4}}}
\right)}
\right)}
\ {dx \sb {1}} \ {dx \sb {2}} \ {dx \sb {3}} \ {dx \sb {4}}\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[\begin{split}{{{P _ {{,4}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {dx _ {4}}}+{{{P _ {{,3}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {dx _ {3}}}+ \\ {{{P _ {{,2}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {dx _ {2}}}+{{{P _ {{,1}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {dx _ {1}}}\end{split}\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[\begin{split}{{{b _ {1}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {{P _ {{,1}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}+ \\ {{{b
_ {2}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {{P _ {{,2}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}+ \\ {{{b
_ {3}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {{P _ {{,3}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}+ \\ {{{b
_ {4}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {{P _ {{,4}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}\end{split}\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[{{{x _ {1}} \over {{{{x _ {2}}} ^ {2}}+{{{x _ {1}}} ^ {2}}}} \ {dx
_ {2}}} -{{{x _ {2}} \over {{{{x _ {2}}} ^ {2}}+{{{x _ {1}}} ^
{2}}}} \ {dx _ {1}}}\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[0\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[s
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[\begin{split}{{{s _ {{,4}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {dx _ {4}}}+{{{s _ {{,3}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {dx _ {3}}}+ \\ {{{s _ {{,2}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {dx _ {2}}}+{{{s _ {{,1}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {dx _ {1}}}\end{split}\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[0\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[\begin{split}{{{s _ {{,1}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {dx _ {2}} \ {dx _ {3}} \ {dx _ {4}}} -{{{s _ {{,2}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {dx _ {1}} \ {dx _ {3}} \ {dx _ {4}}}+ \\ {{{s _ {{,3}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {dx _ {1}} \ {dx _ {2}} \ {dx _ {4}}} -{{{s _ {{,4}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {dx _ {1}} \ {dx _ {2}} \ {dx _ {3}}}\end{split}\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
d (** ( d s)) -- Laplacian(s) dV
\[{\left( {{s \sb {{{,1}{,1}}}}
\left(
{{x \sb {1}}, \: {x \sb {2}}, \: {x \sb {3}}, \: {x \sb {4}}}
\right)}+{{s
\sb {{{,2}{,2}}}}
\left(
{{x \sb {1}}, \: {x \sb {2}}, \: {x \sb {3}}, \: {x \sb {4}}}
\right)}+{{s
\sb {{{,3}{,3}}}}
\left(
{{x \sb {1}}, \: {x \sb {2}}, \: {x \sb {3}}, \: {x \sb {4}}}
\right)}+{{s
\sb {{{,4}{,4}}}}
\left(
{{x \sb {1}}, \: {x \sb {2}}, \: {x \sb {3}}, \: {x \sb {4}}}
\right)}
\right)}
\ {dx \sb {1}} \ {dx \sb {2}} \ {dx \sb {3}} \ {dx \sb {4}}\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[\sin
\left(
{{{x _ {1}} \ {x _ {2}}}}
\right)\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[{{x _ {1}} \ {\cos
\left(
{{{x _ {1}} \ {x _ {2}}}}
\right)}
\ {dx _ {2}}}+{{x _ {2}} \ {\cos
\left(
{{{x _ {1}} \ {x _ {2}}}}
\right)}
\ {dx _ {1}}}\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[{\left( -{{{x _ {2}}} ^ {2}} -{{{x _ {1}}} ^ {2}}
\right)}
\ {\sin
\left(
{{{x _ {1}} \ {x _ {2}}}}
\right)}
\ {dx _ {1}} \ {dx _ {2}} \ {dx _ {3}} \ {dx _ {4}}\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
\[{\left( -{{{x _ {2}}} ^ {2}} -{{{x _ {1}}} ^ {2}}
\right)}
\ {\sin
\left(
{{{x _ {1}} \ {x _ {2}}}}
\right)}\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
** (d (** ( d r)))::EXPR INT
\[{\left( -{{{x _ {2}}} ^ {2}} -{{{x _ {1}}} ^ {2}}
\right)}
\ {\sin
\left(
{{{x _ {1}} \ {x _ {2}}}}
\right)}\]
Type: Expression(Integer)
\[{\left( -{{\pi} ^ {2}} -{{{x _ {2}}} ^ {2}}
\right)}
\ {\sin
\left(
{{{x _ {2}} \ \pi}}
\right)}\]
Type: Expression(Integer)
\[-{{{10} \ {{\pi} ^ {2}} \ {\sin
\left(
{{{{\pi} ^ {2}} \over 3}}
\right)}}
\over 9}\]
Type: Expression(Integer)
\[a
\left(
{{P
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
\right)\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])
d (a(P)*one()$M) -- chain diff
\[\begin{split}{{{P _ {{,4}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {{a _ {{\ }} ^ {,}}
\left(
{{P
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
\right)}
\ {dx _ {4}}}+ \\ {{{P _ {{,3}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {{a _ {{\ }} ^ {,}}
\left(
{{P
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
\right)}
\ {dx _ {3}}}+ \\ {{{P _ {{,2}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {{a _ {{\ }} ^ {,}}
\left(
{{P
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
\right)}
\ {dx _ {2}}}+ \\ {{{P _ {{,1}}}
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}
\ {{a _ {{\ }} ^ {,}}
\left(
{{P
\left(
{{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}}
\right)}}
\right)}
\ {dx _ {1}}}\end{split}\]
Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])