# 4 Usage¶

## 4.0 Examples¶

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

M ==> DFORM(INT,[x,y,z])

$\mathtt{DifferentialForms(Integer,[x,y,z])}$

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

P:=coordVector()$M  $[x,y,z]$ Type: List(Expression(Integer)) and dP:=baseForms()$M

$[dx,dy,dz]$

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

P * dP

$z\ dz + y\ dy + x\ dx$

Type: DeRhamComplex(Integer,[x,y,z]).

There are many ways to create a zero form, one of those is

f := zeroForm(f)$M  $f(x,y,z)$ Type: DeRhamComplex(Integer,[x,y,z]) Now we apply the exterior differential operator to f: d 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 $$M=\mathbb{R}^3$$: T := vectorField(T)$M

$\left[ {{T _ {1}} \left( {x, \: y, \: z} \right)}, \: {{T _ {2}} \left( {x, \: y, \: z} \right)}, \: {{T _ {3}} \left( {x, \: y, \: z} \right)} \right]$

Type: List(Expression(Integer))

Then we build the general one form $$\tau$$:

tau := T * dP


Now we apply the exterior differential operator $$d$$:

d tau

$\begin{split}\scriptstyle{ {{\left( {{{T _ {3}} _ {{,2}}} \left( {x, \: y, \: z} \right)} -{{{T _ {2}} _ {{,3}}} \left( {x, \: y, \: z} \right)} \right)} \ dy \ dz}+{{\left( {{{T _ {3}} _ {{,1}}} \left( {x, \: y, \: z} \right)} -{{{T _ {1}} _ {{,3}}} \left( {x, \: y, \: z} \right)} \right)} \ dx \ dz}+ \\ {{\left( {{{T _ {2}} _ {{,1}}} \left( {x, \: y, \: z} \right)} -{{{T _ {1}} _ {{,2}}} \left( {x, \: y, \: z} \right)} \right)} \ dx \ dy} }\end{split}$

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

d hodgeStar(g, T*dP)$M  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: d 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: %F := hodgeStar(g,F)$M

$\begin{split}\scriptstyle{ {{{B _ {3}} \left( {x, \: y, \: z, \: t} \right)} \ dz \ dt} + {{{B _ {2}} \left( {x, \: y, \: z, \: t} \right)} \ dy \ dt} -{{{E _ {1}} \left( {x, \: y, \: z, \: t} \right)} \ dy \ dz}+ \\ {{{B _ {1}} \left( {x, \: y, \: z, \: t} \right)} \ dx \ dt} + {{{E _ {2}} \left( {x, \: y, \: z, \: t} \right)} \ dx \ dz}- {{{E _ {3}} \left( {x, \: y, \: z, \: t} \right)} \ dx \ dy} }\end{split}$

Type: DeRhamComplex(Integer,[x,y,z,t])

Now, as before:

d %F

$\begin{split}\scriptstyle{ {{\left( -{{{E _ {1}} _ {{,4}}} \left( {x, \: y, \: z, \: t} \right)}+{{{B _ {3}} _ {{,2}}} \left( {x, \: y, \: z, \: t} \right)} -{{{B _ {2}} _ {{,3}}} \left( {x, \: y, \: z, \: t} \right)} \right)} \ dy \ dz \ dt}+ \\ {{\left( {{{E _ {2}} _ {{,4}}} \left( {x, \: y, \: z, \: t} \right)}+{{{B _ {3}} _ {{,1}}} \left( {x, \: y, \: z, \: t} \right)} -{{{B _ {1}} _ {{,3}}} \left( {x, \: y, \: z, \: t} \right)} \right)} \ dx \ dz \ dt}+ \\ {{\left( -{{{E _ {3}} _ {{,4}}} \left( {x, \: y, \: z, \: t} \right)}+{{{B _ {2}} _ {{,1}}} \left( {x, \: y, \: z, \: t} \right)} -{{{B _ {1}} _ {{,2}}} \left( {x, \: y, \: z, \: t} \right)} \right)} \ dx \ dy \ dt}+ \\ {{\left( -{{{E _ {3}} _ {{,3}}} \left( {x, \: y, \: z, \: t} \right)} -{{{E _ {2}} _ {{,2}}} \left( {x, \: y, \: z, \: t} \right)} -{{{E _ {1}} _ {{,1}}} \left( {x, \: y, \: z, \: t} \right)} \right)} \ dx \ dy \ dz} }\end{split}$

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 $$\langle F,F\rangle$$:

dot(g,F,F)$M  $\begin{split}\scriptstyle{ -{{{{E _ {3}} \left( {x, \: y, \: z, \: t} \right)}} ^ {2}} -{{{{E _ {2}} \left( {x, \: y, \: z, \: t} \right)}} ^ {2}} -{{{{E _ {1}} \left( {x, \: y, \: z, \: t} \right)}} ^ {2}}+ \\ {{{{B _ {3}} \left( {x, \: y, \: z, \: t} \right)}} ^ {2}}+{{{{B _ {2}} \left( {x, \: y, \: z, \: t} \right)}} ^ {2}}+{{{{B _ {1}} \left( {x, \: y, \: z, \: t} \right)}} ^ {2}} }\end{split}$ Type: Expression(Integer) and $$\langle F,\star F\rangle$$: dot(g,F,%F)$M

$\scriptstyle{ -{2 \ {{B _ {3}} \left( {x, \: y, \: z, \: t} \right)} \ {{E _ {3}} \left( {x, \: y, \: z, \: t} \right)}} -{2 \ {{B _ {2}} \left( {x, \: y, \: z, \: t} \right)} \ {{E _ {2}} \left( {x, \: y, \: z, \: t} \right)}} -{2 \ {{B _ {1}} \left( {x, \: y, \: z, \: t} \right)} \ {{E _ {1}} \left( {x, \: y, \: z, \: t} \right)}} }$

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

### 4.3 Some Examples from Maple¶

Examples from Maple.

#### 4.3.1 5-dimensional Manifold¶

First create a 5-dimensional manifold M and define a metric tensor g on the tangent space of M:

v:=[x[j] for j in 1..5]
M:=DFORM(INT,v)
g:=diagonalMatrix([1,1,1,1,1])::SquareMatrix(5,INT)
dX:=baseForms()$M  hodgeStar(g,dX.1)$M

${dx _ {2}} \ {dx _ {3}} \ {dx _ {4}} \ {dx _ {5}}$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4],x[5]])

hodgeStar(g,dX.2)$M  $-{{dx _ {1}} \ {dx _ {3}} \ {dx _ {4}} \ {dx _ {5}}}$ Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4],x[5]]) hodgeStar(g,dX.2*dX.3)$M

${dx _ {1}} \ {dx _ {4}} \ {dx _ {5}}$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4],x[5]])

hodgeStar(g,dX.2*dX.4)$M  $-{{dx _ {1}} \ {dx _ {3}} \ {dx _ {5}}}$ Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4],x[5]]) hodgeStar(g,dX.2*dX.3*dX.4)$M

$-{{dx _ {1}} \ {dx _ {5}}}$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4],x[5]])

We see an exact match with the published results.

#### 4.3.2 General metric (2-dim)¶

To show the dependence of the Hodge star operator upon the metric, we consider a general metric g on a 2-dimensional manifold.

v:=[x,y]
M:=DFORM(INT,v)
R ==> EXPR INT
g:=matrix([[a::R,b],[b,c]])::SquareMatrix(2,R)
[dx,dy]:=baseForms()$M  hodgeStar(g,dx)$M

${{{c \ {\sqrt {{abs \left( {{{a \ c} -{{b} ^ {2}}}} \right)}}}} \over {{a \ c} -{{b} ^ {2}}}} \ dy}+{{{b \ {\sqrt {{abs \left( {{{a \ c} -{{b} ^ {2}}}} \right)}}}} \over {{a \ c} -{{b} ^ {2}}}} \ dx}$

Type: DeRhamComplex(Integer,[x,y])

hodgeStar(g,dy)$M  $-{{{b \ {\sqrt {{abs \left( {{{a \ c} -{{b} ^ {2}}}} \right)}}}} \over {{a \ c} -{{b} ^ {2}}}} \ dy} -{{{a \ {\sqrt {{abs \left( {{{a \ c} -{{b} ^ {2}}}} \right)}}}} \over {{a \ c} -{{b} ^ {2}}}} \ dx}$ Type: DeRhamComplex(Integer,[x,y]) f := hodgeStar(g,dx*dy)$M

${\sqrt {{abs \left( {{{a \ c} -{{b} ^ {2}}}} \right)}}} \over {{a \ c} -{{b} ^ {2}}}$

Type: DeRhamComplex(Integer,[x,y])

hodgeStar(g,f)$M  ${{abs \left( {{{a \ c} -{{b} ^ {2}}}} \right)} \over {{a \ c} -{{b} ^ {2}}}} \ dx \ dy$ Type: DeRhamComplex(Integer,[x,y]) #### 4.3.3 Laplacian¶ The Laplacian of a function with respect to a metric g can be calculated using the exterior derivative and the Hodge star operator. Generally, the following identity holds: $\Delta = d \circ \delta + \delta \circ d$ where $$\delta:=(-1)^p\, \star^{-1}\,d \,\star$$ is the codifferential to be applied on a p-form (resulting in a (p-1)-form). Therefore, the Laplacian applied to a function f (zero form) is: $\Delta f = \delta \circ df = \star^{-1}\, d \,\star df= \star\, d \, \star df.$ v:=[r,u] -- polar coordinates M:=DFORM(INT,v) R ==> EXPR INT g:=matrix([[1,0],[0,r^2]])::SquareMatrix(2,R) [dr,du]:=baseForms()$M


A function on M can easiliy be defined by

f:=zeroForm(f)$M  $f\left({r, \: u}\right)$ Type: DeRhamComplex(Integer,[r,u]) We translate the formula: hodgeStar(g, d hodgeStar(g,d f)$M)$M  ${{{abs \left( {{{r} ^ {2}}} \right)} \ {{f _ {{{,2}{,2}}}} \left( {r, \: u} \right)}}+{{{r} ^ {2}} \ {abs \left( {{{r} ^ {2}}} \right)} \ {{f _ {{{,1}{,1}}}} \left( {r, \: u} \right)}}+{r \ {abs \left( {{{r} ^ {2}}} \right)} \ {{f _ {{,1}}} \left( {r, \: u} \right)}}} \over {{r} ^ {4}}$ Type: DeRhamComplex(Integer,[r,u]) Simplifying yields for M: $\Delta_M f = \frac{\partial^2 f}{\partial r^2} + \frac{1}{r} \frac{\partial f}{\partial r} + \frac{1}{r^2} \frac{\partial^2 f}{\partial u^2}$ #### 4.3.4 Lie derivative¶ v:=[x[i] for i in 1..3] M:=DFORM(INT,v) dX:=baseForms()$M
V:=vectorField(V)$M f:=scalarField(f)$M

lieDerivative(V,dX.1)

$\scriptstyle{ {{{{V _ {1}} _ {{,3}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}} \right)} \ {dx _ {3}}}+{{{{V _ {1}} _ {{,2}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}} \right)} \ {dx _ {2}}}+{{{{V _ {1}} _ {{,1}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}} \right)} \ {dx _ {1}}}}$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3]])

lieDerivative(V,f*dX.1)

$\cal{L}_V\,f\,dx_1$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3]])

lieDerivative(V,f*dX.1*dX.2)

$\cal{L}_V\,f\,dx_1\wedge dx_2$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3]])

### 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 ** !

w:=x.1*dx.2-x.2*dx.1

${{x _ {1}} \ {dx _ {2}}} -{{x _ {2}} \ {dx _ {1}}}$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])

d w

$2 \ {dx _ {1}} \ {dx _ {2}}$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])

w*w

$0$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])

i(vf,w)

${{x _ {1}} \ {{b _ {2}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}} -{{x _ {2}} \ {{b _ {1}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}}$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])

L(vf,w)

$\begin{split}{{\left( {{x _ {1}} \ {{{b _ {2}} _ {{,4}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}} -{{x _ {2}} \ {{{b _ {1}} _ {{,4}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}} \right)} \ {dx _ {4}}}+ \\ {{\left( {{x _ {1}} \ {{{b _ {2}} _ {{,3}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}} -{{x _ {2}} \ {{{b _ {1}} _ {{,3}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}} \right)} \ {dx _ {3}}}+ \\ {{\left( {{x _ {1}} \ {{{b _ {2}} _ {{,2}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}} -{{x _ {2}} \ {{{b _ {1}} _ {{,2}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}}+{{b _ {1}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \right)} \ {dx _ {2}}}+ \\ {{\left( {{x _ {1}} \ {{{b _ {2}} _ {{,1}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}} -{{x _ {2}} \ {{{b _ {1}} _ {{,1}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}} -{{b _ {2}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \right)} \ {dx _ {1}}}\end{split}$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])

d i(vf,w) + i(vf,d w)

$\begin{split}{{\left( {{x _ {1}} \ {{{b _ {2}} _ {{,4}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}} -{{x _ {2}} \ {{{b _ {1}} _ {{,4}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}} \right)} \ {dx _ {4}}}+ \\ {{\left( {{x _ {1}} \ {{{b _ {2}} _ {{,3}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}} -{{x _ {2}} \ {{{b _ {1}} _ {{,3}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}} \right)} \ {dx _ {3}}}+ \\ {{\left( {{x _ {1}} \ {{{b _ {2}} _ {{,2}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}} -{{x _ {2}} \ {{{b _ {1}} _ {{,2}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}}+{{b _ {1}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \right)} \ {dx _ {2}}}+ \\ {{\left( {{x _ {1}} \ {{{b _ {2}} _ {{,1}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}} -{{x _ {2}} \ {{{b _ {1}} _ {{,1}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}} -{{b _ {2}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \right)} \ {dx _ {1}}}\end{split}$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])

% - L(vf,w)

$0$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])

dot(g,w,w)$M  ${{{x _ {2}}} ^ {2}}+{{{x _ {1}}} ^ {2}}$ 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]]) d (P*one()$M) -- One()?

$\begin{split}{{{P _ {{,4}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \ {dx _ {4}}}+{{{P _ {{,3}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \ {dx _ {3}}}+ \\ {{{P _ {{,2}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \ {dx _ {2}}}+{{{P _ {{,1}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \ {dx _ {1}}}\end{split}$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])

i(vf,%)

$\begin{split}{{{b _ {1}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \ {{P _ {{,1}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}}+ \\ {{{b _ {2}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \ {{P _ {{,2}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}}+ \\ {{{b _ {3}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \ {{P _ {{,3}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}}+ \\ {{{b _ {4}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \ {{P _ {{,4}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}}\end{split}$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])

1/dot(g,w,w)$M*w  ${{{x _ {1}} \over {{{{x _ {2}}} ^ {2}}+{{{x _ {1}}} ^ {2}}}} \ {dx _ {2}}} -{{{x _ {2}} \over {{{{x _ {2}}} ^ {2}}+{{{x _ {1}}} ^ {2}}}} \ {dx _ {1}}}$ Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]]) d %  $0$ Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]]) s:=zeroForm('s)$M

$s \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])

d s

$\begin{split}{{{s _ {{,4}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \ {dx _ {4}}}+{{{s _ {{,3}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \ {dx _ {3}}}+ \\ {{{s _ {{,2}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \ {dx _ {2}}}+{{{s _ {{,1}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \ {dx _ {1}}}\end{split}$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])

d (** s)

$0$

Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]])

** ( d s)

$\begin{split}{{{s _ {{,1}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \ {dx _ {2}} \ {dx _ {3}} \ {dx _ {4}}} -{{{s _ {{,2}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \ {dx _ {1}} \ {dx _ {3}} \ {dx _ {4}}}+ \\ {{{s _ {{,3}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \ {dx _ {1}} \ {dx _ {2}} \ {dx _ {4}}} -{{{s _ {{,4}}} \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)} \ {dx _ {1}} \ {dx _ {2}} \ {dx _ {3}}}\end{split}$

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]])

r:=sin(x.1*x.2)*one()$M  $\sin \left( {{{x _ {1}} \ {x _ {2}}}} \right)$ Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]]) d r  ${{x _ {1}} \ {\cos \left( {{{x _ {1}} \ {x _ {2}}}} \right)} \ {dx _ {2}}}+{{x _ {2}} \ {\cos \left( {{{x _ {1}} \ {x _ {2}}}} \right)} \ {dx _ {1}}}$ Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]]) d (** ( d r))  ${\left( -{{{x _ {2}}} ^ {2}} -{{{x _ {1}}} ^ {2}} \right)} \ {\sin \left( {{{x _ {1}} \ {x _ {2}}}} \right)} \ {dx _ {1}} \ {dx _ {2}} \ {dx _ {3}} \ {dx _ {4}}$ Type: DeRhamComplex(Integer,[x[1],x[2],x[3],x[4]]) ** (d (** ( d r)))  ${\left( -{{{x _ {2}}} ^ {2}} -{{{x _ {1}}} ^ {2}} \right)} \ {\sin \left( {{{x _ {1}} \ {x _ {2}}}} \right)}$ 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) eval(%,xs.1=%pi)  ${\left( -{{\pi} ^ {2}} -{{{x _ {2}}} ^ {2}} \right)} \ {\sin \left( {{{x _ {2}} \ \pi}} \right)}$ Type: Expression(Integer) eval(%,xs.2=%pi/3)  $-{{{10} \ {{\pi} ^ {2}} \ {\sin \left( {{{{\pi} ^ {2}} \over 3}} \right)}} \over 9}$ Type: Expression(Integer) a(P)*one()$M

$a \left( {{P \left( {{x _ {1}}, \: {x _ {2}}, \: {x _ {3}}, \: {x _ {4}}} \right)}} \right)$

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]])