Ordinary differential calculus may be formalized and generalized to arbitrary-dimensional oriented manifolds using the exterior calculus. Here I show how the stokes package furnishes functionality for working with the exterior calculus, and provide numerical verification of a number of theorems. Notation follows that of Spivak, and Hubbard and Hubbard.
Recall that a \(k\)-tensor is a multilinear map \(S\colon V^k\longrightarrow\mathbb{R}\), where \(V=\mathbb{R}^n\) is considered as a vector space; Spivak denotes the space of multilinear maps as \(\mathcal{J}^k(V)\). Formally, multilinearity means
\[ S\left(v_1,\ldots,av_i,\ldots,v_k\right)=a\cdot S\left(v_1,\ldots,v_i,\ldots,v_k\right) \]
and
\[S\left(v_1,\ldots,v_i+{v_i}',\ldots,v_k\right)=S\left(v_1,\ldots,v_i,\ldots,x_v\right)+ S\left(v_1,\ldots,{v_i}',\ldots,v_k\right). \]
where \(v_i\in V\). If \(S\in\mathcal{J}^k(V)\) and \(T\in\mathcal{J}^l(V)\), then we may define \(S\otimes T\in\mathcal{J}^{k+l}(V)\) as
\[ S\otimes T\left(v_1,\ldots,v_k,v_{k+1},\ldots,v_{k+l}\right)= S\left(v_1,\ldots,v_k\right)\cdot T\left(v_1,\ldots,v_l\right) \]
Spivak observes that \(\mathcal{J}^k(V)\) is spanned by the \(n^k\) products of the form
\[ \phi_{i_1}\otimes\phi_{i_2}\otimes\cdots\otimes\phi_{i_k}\qquad 1\leq i_i,i_2,\ldots,i_k\leq n \]
where \(v_1,\ldots,v_k\) is a basis for \(V\) and \(\phi_i\left(v_j\right)=\delta_{ij}\); we can therefore write
\[ S=\sum_{1\leq i_1,\ldots,i_k\leq n} a_{i_1\ldots i_k} \phi_{i_1}\otimes\cdots\otimes\phi_{i_k}. \]
The space spanned by such products has a natural representation in R as an array of dimensions \(n\times\cdots\times n=n^k\). If A is such an array, then the element A[i_1,i_2,...,i_k] is the coefficient of \(\phi_{i_1}\otimes\ldots\otimes\phi_{i_k}\). However, it is more efficient and conceptually cleaner to consider a sparse array, as implemented by the spray package. We will consider the case \(n=5,k=4\), so we have multilinear maps from \(\left(\mathbb{R}^5\right)^4\) to \(\mathbb{R}\). Below, we will test algebraic identities in R using the idiom furnished by the stokes package. For our example we will define \(S=1.5\phi_5\otimes\phi_1\otimes\phi_1\otimes\phi_1+2.5\phi_1\otimes\phi_1\otimes\phi_2\otimes\phi_3+3.5\phi_1\otimes\phi_3\otimes\phi_4\otimes\phi_2\) using a matrix with three rows, one per term, and whose rows correspond to each term’s cross products of the \(\phi\)’s. We first have to load the stokes package:
Then the idiom is straightforward:
## [,1] [,2] [,3] [,4]
## [1,] 5 1 1 1
## [2,] 1 1 2 3
## [3,] 1 3 4 2## A linear map from V^4 to R with V=R^5:
## val
## 1 3 4 2 = 3.5
## 1 1 2 3 = 2.5
## 5 1 1 1 = 1.5Observe that, if stored as an array of size \(n^k\), \(S\) would have \(5^4=625\) elements, all but three of which are zero. So \(S\) is a 4-tensor, mapping \(V^4\) to \(\mathbb{R}\), where \(V=\mathbb{R}^5\). Here we have \(S=1.5\phi_5\otimes\phi_1\otimes\phi_1\otimes\phi_1+2.5\phi_1\otimes\phi_1\otimes\phi_2\otimes\phi_3+3.5\phi_1\otimes\phi_3\otimes\phi_4\otimes\phi_2\). Note that in some implementations the row order of object S will differ from that of M; this phenomenon is due to the underlying C implementation using the STL map class and is discussed in more detail in the mvp package.
First, we will define \(E\) to be a random point in \(V^k\) in terms of a matrix:
## [,1] [,2] [,3] [,4]
## [1,] 1.2629543 -1.539950042 0.7635935 -0.4115108
## [2,] -0.3262334 -0.928567035 -0.7990092 0.2522234
## [3,] 1.3297993 -0.294720447 -1.1476570 -0.8919211
## [4,] 1.2724293 -0.005767173 -0.2894616 0.4356833
## [5,] 0.4146414 2.404653389 -0.2992151 -1.2375384Recall that \(n=5\), \(k=4\), so \(E\in\left(\mathbb{R}^5\right)^4\). We can evaluate \(S\) at \(E\) as follows:
## [1] -3.068997Tensors have a natural vector space structure; they may be added and subtracted, and multiplied by a scalar, the same as any other vector space. Below, we define a new tensor \(S_1\) and work with \(2S-3S_1\):
## A linear map from V^4 to R with V=R^5:
## val
## 1 3 4 2 = 7
## 1 1 2 3 = 5
## 5 1 1 1 = 3
## 1 1 1 2 = -12
## 1 1 2 1 = -9
## 1 2 1 1 = -6
## 2 1 1 1 = -3We may verify that tensors are linear using package idiom:
way1 <- as.function(2*S-3*S1)(E)
way2 <- 2*as.function(S)(E) -3*as.function(S1)(E)
c(way1,way2,way1-way2)## [1] 2.374816e+00 2.374816e+00 -4.440892e-16(that is, identical up to numerical precision).
Testing multilinearity is straightforward in the package. To do this, we need to define three matrices E1,E2,E3 corresponding to points in \(\left(\mathbb{R}^5\right)^4\) which are identical except for one column. In E3, this column is a linear combination of the corresponding column in E2 and E3:
E1 <- E
E2 <- E
E3 <- E
x1 <- rnorm(n)
x2 <- rnorm(n)
r1 <- rnorm(1)
r2 <- rnorm(1)
E1[,2] <- x1
E2[,2] <- x2
E3[,2] <- r1*x1 + r2*x2Then we can verify the multilinearity of \(S\) by coercing to a function which is applied to E1, E2, E3:
## [1] -0.5640577 -0.5640577 0.0000000(that is, identical up to numerical precision). Note that this is not equivalent to linearity over \(V^{nk}\):
E1 <- matrix(rnorm(n*k),n,k)
E2 <- matrix(rnorm(n*k),n,k)
way1 <- f(r1*E1+r2*E2)
way2 <- r1*f(E1)+r2*f(E2)
c(way1,way2,way1-way2)## [1] 0.1731245 0.3074186 -0.1342941Given two k-tensor objects \(S,T\) we can form the cross product \(S\otimes T\), defined as
\[ S\otimes T\left(v_1,\ldots,v_k,v_{k+1},\ldots, v_{k+l}\right)= S\left(v_1,\ldots v_k\right)\cdot T\left(v_{k+1},\ldots v_{k+l}\right) \]
We will calculate the cross product of two tensors S1,S2 defined as follows:
## A linear map from V^2 to R with V=R^4:
## val
## 3 4 = 3
## 2 3 = 2
## 1 2 = 1## A linear map from V^3 to R with V=R^6:
## val
## 2 4 6 = 1
## 1 3 5 = 1The R idiom for \(S1\otimes S2\) would be cross(), or %X%:
## A linear map from V^5 to R with V=R^6:
## val
## 1 2 1 3 5 = 1
## 3 4 1 3 5 = 3
## 1 2 2 4 6 = 1
## 2 3 2 4 6 = 2
## 2 3 1 3 5 = 2
## 3 4 2 4 6 = 3Then, for example:
E <- matrix(rnorm(30),6,5)
way1 <- as.function(cross(S1,S2))(E)
way2 <- as.function(S1)(E) * as.function(S2)(E)
c(way2,way2,way1-way2)## [1] 2.00319 2.00319 -3.05152(that is, identical up to numerical precision).
An alternating form is a multilinear map \(T\) satisfying
\[ \mathrm{T}\left(v_1,\ldots,v_i,\ldots,v_j,\ldots,v_k\right)= -\mathrm{T}\left(v_1,\ldots,v_j,\ldots,v_i,\ldots,v_k\right) \]
(or, equivalently, \(\mathrm{T}\left(v_1,\ldots,v_i,\ldots,v_i,\ldots,v_k\right)= 0\)). We write \(\Lambda^k(V)\) for the space of all alternating multilinear maps from \(V^k\) to \(\mathbb{R}\). Spivak gives \(\operatorname{Alt}\colon\mathcal{J}^k(V)\longrightarrow\Lambda^k(V)\) defined by
\[\operatorname{Alt}(T)\left(v_1,\ldots,v_k\right)= \frac{1}{k!}\sum_{\sigma\in S_k}\operatorname{sgn}(\sigma)\cdot T\left(v_{\sigma(1)},\ldots,v_{\sigma(k)}\right) \]
where the sum ranges over all permutations of \(\left[n\right]=\left\{1,2,\ldots,n\right\}\) and \(\operatorname{sgn}(\sigma)\in\pm 1\) is the sign of the permutation. If \(T\in\mathcal{J}^k(V)\) and \(\omega\in\Lambda^k(V)\), it is straightforward to prove that \(\operatorname{Alt}(T)\in\Lambda^k(V)\), \(\operatorname{Alt}\left(\operatorname{Alt}\left(T\right)\right)=\operatorname{Alt}\left(T\right)\), and \(\operatorname{Alt}\left(\omega\right)=\omega\).
In the stokes package, this is effected by the Alt() function:
## A linear map from V^2 to R with V=R^4:
## val
## 3 4 = 3
## 2 3 = 2
## 1 2 = 1## A linear map from V^2 to R with V=R^4:
## val
## 4 3 = -1.5
## 3 2 = -1.0
## 2 3 = 1.0
## 3 4 = 1.5
## 2 1 = -0.5
## 1 2 = 0.5Verifying that S1 is in fact alternating is straightforward:
E <- matrix(rnorm(8),4,2)
Erev <- E[,2:1]
as.function(Alt(S1))(E) + as.function(Alt(S1))(Erev) # should be zero## [1] 0However, we can see that this form for alternating tensors (here called \(k\)-forms) is inefficient and highly redundant: in this example there is a 1 2 term and a 2 1 term (the coefficients are equal and opposite). In this example we have \(k=2\) but in general there would be potentially \(k!\) essentially repeated terms which collectively require only a single coefficient. The package provides kform objects which are inherently alternating using a more efficient representation; they are described using wedge products which are discussed next.
This section follows the exposition of Hubbard and Hubbard, who introduce the exterior calculus starting with a discussion of elementary forms, which are alternating forms with a particularly simple structure. An example of an elementary form would be \(dx_1\wedge dx_3\) [treated as an indivisible entity], which is an alternating multilinear map from \(\mathbb{R}^n\times\mathbb{R}^n\) to \(\mathbb{R}\) with
\[ \left( dx_1\wedge dx_3 \right)\left( \begin{pmatrix}a_1\\a_2\\a_3\\ \vdots\\ a_n\end{pmatrix}, \begin{pmatrix}b_1\\b_3\\b_3\\ \vdots\\ b_n\end{pmatrix} \right)=\mathrm{det} \begin{pmatrix} a_1 & b_1 \\ a_3 & b_3\end{pmatrix} =a_1b_3-a_3b_1 \]
That this is alternating follows from the properties of the determinant. In general of course, \(dx_i\wedge dx_j\left( \begin{pmatrix}a_1\\ \vdots\\ a_n\end{pmatrix}, \begin{pmatrix}b_1\\ \vdots\\ b_n\end{pmatrix} \right)=\mathrm{det} \begin{pmatrix} a_i & b_i \\ a_j & b_j\end{pmatrix}\). Because such objects are linear, it is possible to consider sums of elementary forms, such as \(dx_1\wedge dx_2 + 3 dx_2\wedge dx_3\) with
\[ \left( dx_1\wedge dx_2 + 3dx_2\wedge dx_3 \right)\left( \begin{pmatrix}a_1\\a_2\\ \vdots\\ a_n\end{pmatrix}, \begin{pmatrix}b_1\\b_2\\ \vdots\\ b_n\end{pmatrix} \right)=\mathrm{det} \begin{pmatrix} a_1 & b_1\\ a_2 & b_2\end{pmatrix} +3\mathrm{det} \begin{pmatrix} a_2 & b_2\\ a_3 & b_3\end{pmatrix} \]
or even \(K=dx_1\wedge dx_2\wedge dx_3 +5dx_1\wedge dx_2\wedge dx_4\) which would be a linear map from \(\left(\mathbb{R}^n\right)^3\) to \(\mathbb{R}\) with
\[ \left( dx_4\wedge dx_2\wedge dx_3 +5dx_1\wedge dx_2\wedge dx_4 \right)\left( \begin{pmatrix}a_1\\a_2\\ \vdots\\ a_n\end{pmatrix}, \begin{pmatrix}b_1\\b_2\\ \vdots\\ b_n\end{pmatrix}, \begin{pmatrix}c_1\\c_2\\ \vdots\\ c_n\end{pmatrix} \right)=\mathrm{det} \begin{pmatrix} a_4 & b_4 & c_4\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{pmatrix} +5\mathrm{det} \begin{pmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_4 & b_4 & c_4 \end{pmatrix}. \]
Defining \(K\) has ready R idiom in which we define a matrix whose rows correspond to the differentials in each term:
## [,1] [,2] [,3]
## [1,] 4 2 3
## [2,] 1 4 2## An alternating linear map from V^3 to R with V=R^4:
## -5 dt^dx^dz + dx^dy^dzFunction as.kform() takes each row of M and places the elements in increasing order; the coefficient will change sign if the permutation is odd. Note that the order of the rows in K is immaterial and indeed in some implementations will appear in a different order: the stokes package uses the spray package, which in turn utilises the STL map class of C++.
In the previous section we defined objects such as “\(dx_1\wedge dx_6\)” as a single entity. Here I define the elementary form \(dx_i\) formally and in the next section discuss the wedge product \(\wedge\). The elementary form \(dx_i\) is simply a map from \(\mathbb{R}^n\) to \(\mathbb{R}\) with \(dx_i\left(x_1,x_2,\ldots,x_n\right)=x_i\). Observe that \(dx_i\) is an alternating form, even though we cannot swap arguments (because there is only one). Package idiom for creating an elementary form appears somewhat cryptic at first sight, but is consistent (it is easier to understand package idiom for creating more complicated alternating forms, as in the next section). Suppose we wish to work with \(dx_3\):
dx3 <- as.kform(matrix(3,1,1),1)
options(kform_symbolic_print = NULL) # revert to default print method
dx3## An alternating linear map from V^1 to R with V=R^3:
## val
## 3 = 1Interpretation of the output above is not obvious (it is easier to understand the output from more complicated alternating forms, as in the next section), but for the moment observe that \(dx_3\) is indeed an alternating form, mapping \(\mathbb{R}^n\) to \(\mathbb{R}\) with \(dx_3\left(x_1,x_2,\ldots,x_n\right)=x_3\). Thus, for example:
## [1] 16## [1] 16and we see that \(dx_3\) picks out the third element of a vector. These are linear in the sense that we may add and subtract these elementary forms:
## [1] 13The wedge product maps two alternating forms to another alternating form; given \(\omega\in\Lambda^k(V)\) and \(\eta\in\Lambda^l(V)\), Spivak defines the wedge product \(\omega\wedge\eta\in\Lambda^{k+l}(V)\) as
\[ \omega\wedge\eta={k+l\choose k\quad l}\operatorname{Alt}(\omega\otimes\eta) \]
and this is implemented in the package by function wedge(), or, more idiomatically, ^:
## An alternating linear map from V^3 to R with V=R^6:
## val
## 1 4 6 = 7
## 3 4 5 = -2## An alternating linear map from V^2 to R with V=R^7:
## val
## 5 7 = 5
## 4 6 = 4
## 3 5 = 3
## 2 4 = 2
## 1 3 = 1In symbolic notation, K1 is equal to \(7dx_1\wedge dx_4\wedge dx_6 -2dx_3\wedge dx_4\wedge dx_5\). and K2 is \(dx_1\wedge dx_3+ 2dx_2\wedge dx_4+ 3dx_3\wedge dx_5+ 4dx_4\wedge dx_6+ 5dx_5\wedge dx_7\). Package idiom for wedge products is straightforward:
## An alternating linear map from V^5 to R with V=R^7:
## val
## 1 3 4 5 6 = -21
## 1 4 5 6 7 = -35(we might write the product as \(-35dx_1\wedge dx_4\wedge dx_5\wedge dx_6\wedge dx_7 -21dx_1\wedge dx_3\wedge dx_4\wedge dx_5\wedge dx_6\)). See how the wedge product eliminates rows with repeated entries, gathers permuted rows together (respecting the sign of the permutation), and expresses the result in terms of elementary forms. The product is a linear combination of two elementary forms; note that only two coefficients out of a possible \({7\choose 5}=21\) are nonzero. Note again that the order of the rows in the product is arbitrary.
The wedge product has formal properties such as distributivity but by far the most interesting one is associativity, which I will demonstrate below:
F1 <- as.kform(matrix(c(3,4,5, 4,6,1,3,2,1),3,3,byrow=TRUE))
F2 <- as.kform(cbind(1:6,3:8),1:6)
F3 <- kform_general(1:8,2)
(F1 ^ F2) ^ F3## An alternating linear map from V^7 to R with V=R^8:
## val
## 1 2 4 5 6 7 8 = -5
## 2 3 4 5 6 7 8 = 6
## 1 2 3 5 6 7 8 = 11
## 1 2 3 4 5 6 8 = 1
## 1 2 3 4 5 7 8 = -5
## 1 2 3 4 6 7 8 = 2
## 1 2 3 4 5 6 7 = 1
## 1 3 4 5 6 7 8 = -2## An alternating linear map from V^7 to R with V=R^8:
## val
## 2 3 4 5 6 7 8 = 6
## 1 2 4 5 6 7 8 = -5
## 1 2 3 4 5 7 8 = -5
## 1 2 3 4 5 6 8 = 1
## 1 2 3 4 6 7 8 = 2
## 1 2 3 4 5 6 7 = 1
## 1 3 4 5 6 7 8 = -2
## 1 2 3 5 6 7 8 = 11Note carefully in the above that the terms in (F1 ^ F2) ^ F3 and F1 ^ (F2 ^ F3) appear in a different order. They are nevertheless algebraically identical, as we may demonstrate by calculating their difference:
## The zero alternating linear map from V^7 to R with V=R^n:
## empty sparse array with 7 columnsSpivak observes that \(\Lambda^k(V)\) is spanned by the \(n\choose k\) wedge products of the form
\[ dx_{i_1}\wedge dx_{i_2}\wedge\ldots\wedge dx_{i_k}\qquad 1\leq i_i<i_2<\cdots <i_k\leq n \]
where these products are the elementary forms (compare \(\mathcal{J}^k(V)\), which is spanned by \(n^k\) elementary forms). Formally, multilinearity means every element of the space \(\Lambda^k(V)\) is a linear combination of elementary forms, as illustrated in the package by function kform_general(). Consider the following idiom:
## An alternating linear map from V^2 to R with V=R^4:
## val
## 3 4 = 6
## 2 4 = 5
## 1 4 = 4
## 2 3 = 3
## 1 3 = 2
## 1 2 = 1Object Krel is a two-form, specifically a map from \(\left(\mathbb{R}^4\right)^2\) to \(\mathbb{R}\). Observe that Krel has \({4\choose 2}=6\) components, which do not appear in any particular order. Addition of such \(k\)-forms is straightforward in R idiom but algebraically nontrivial:
K1 <- as.kform(matrix(1:4,2,2),c(1,109))
K2 <- as.kform(matrix(c(1,3,7,8,2,4),ncol=2,byrow=TRUE),c(-1,5,4))
K1## An alternating linear map from V^2 to R with V=R^4:
## val
## 2 4 = 109
## 1 3 = 1## An alternating linear map from V^2 to R with V=R^8:
## val
## 2 4 = 4
## 7 8 = 5
## 1 3 = -1## An alternating linear map from V^2 to R with V=R^8:
## val
## 2 4 = 113
## 7 8 = 5In the above, note how the \(dx_2\wedge dx_4\) terms combine [to give 2 4 = 113] and the \(dx_1\wedge dx_3\) term vanishes by cancellation.
Although the spray form used above is probably the most direct and natural representation of differential forms in numerical work, sometimes we need a more algebraic print method.
## A linear map from V^2 to R with V=R^5:
## val
## 4 5 = 4
## 3 4 = 3
## 2 3 = 2
## 1 2 = 1we can represent this more algebraically using the as.symbolic() function:
## [1] +4 d*e +3 c*d +2 b*c + a*bIn the above, U is a multilinear map from \(\left(\mathbb{R}^5\right)^2\) to \(\mathbb{R}\). Symbolically, a represents the map that takes \((a,b,c,d,e)\) to \(a\), b the map that takes \((a,b,c,d,e)\) to b, and so on. The asterisk * represents the tensor product \(\otimes\). Alternating forms work similarly but \(k\)-forms have different defaults:
## An alternating linear map from V^2 to R with V=R^3:
## val
## 2 3 = 3
## 1 3 = 2
## 1 2 = 1## [1] +3 dx^dy +2 dw^dy + dw^dxNote that the wedge product \(\wedge\), although implemented in package idiom as ^ or %^%, appears in the symbolic representation as an ascii caret, ^.
We can alter the default print method with the kform_symbolic_print option, which uses as.symbolic():
## An alternating linear map from V^2 to R with V=R^3:
## +3 dx2^dx3 +2 dx1^dx3 + dx1^dx2This print option works nicely with the d() function for elementary forms:
## An alternating linear map from V^3 to R with V=R^7:
## - dx3^dx5^dx7 + dx1^dx3^dx7 +5 dx2^dx5^dx7 -5 dx1^dx2^dx7Given a \(k\)-form \(\phi\colon V^k\longrightarrow\mathbb{R}\) and a vector \(\mathbf{v}\in V\), the contraction \(\phi_\mathbf{v}\) of \(\phi\) and \(\mathbf{v}\) is a \(k-1\)-form with
\[ \phi_\mathbf{v}\left(\mathbf{v}^1,\ldots,\mathbf{v}^{k-1}\right) = \phi\left(\mathbf{v},\mathbf{v}^1,\ldots,\mathbf{v}^{k-1}\right) \]
if \(k>1\); we specify \(\phi_\mathbf{v}=\phi(\mathbf{v})\) if \(k=1\). Verification is straightforward:
## An alternating linear map from V^3 to R with V=R^7:
## val
## 2 4 6 = -6
## 4 6 7 = 5
## 5 6 7 = 4
## 1 3 7 = 7
## 1 5 7 = -12
## 1 4 6 = 8
## 2 3 7 = -2
## 1 2 4 = 1V <- matrix(runif(21),ncol=3)
LHS <- as.function(o)(V)
RHS <- as.function(contract(o,V[,1]))(V[,-1])
c(LHS=LHS,RHS=RHS,diff=LHS-RHS)## LHS RHS diff
## 4.512547e-01 4.512547e-01 -5.551115e-17It is possible to iterate the contraction process; if we pass a matrix \(V\) to contract() then this is interpreted as repeated contraction with the columns of \(V\):
## [1] 0.4512547If we pass three columns to contract() the result is a \(0\)-form:
## [1] 0.4512547In the above, the result is coerced to a scalar; in order to work with a formal \(0\)-form (which is represented in the package as a spray with a zero-column index matrix) we can use the lose=FALSE argument:
## An alternating linear map from V^0 to R with V=R^0:
## val
## = 0.4512547Suppose we are given a two-form \(\omega=\sum_{i<j}a_{ij}dx_i\wedge dx_j\) and relationships \(dx_i=\sum_rM_{ir}dy_r\), then we would have
\[ \omega = \sum_{i<j} a_{ij}\left(\sum_rM_{ir}dy_r\right)\wedge\left(\sum_rM_{jr}dy_r\right). \]
The general situation would be a \(k\)-form where we would have
\[ \omega=\sum_{i_1<\cdots<i_k}a_{i_1\ldots i_k}dx_{i_1}\wedge\cdots\wedge dx_{i_k} \]
giving
\[\omega = \sum_{i_1<\cdots <i_k}\left[ a_{i_1<\cdots < i_k}\left(\sum_rM_{i_1r}dy_r\right)\wedge\cdots\wedge\left(\sum_rM_{i_kr}dy_r\right)\right]. \]
So \(\omega\) was given in terms of \(dx_1,\ldots,dx_k\) and we have expressed it in terms of \(dy_1,\ldots,dy_k\). So for example if
\[ \omega= dx_1\wedge dx_2 + 5dx_1\wedge dx_3\]
and
\[ \left( \begin{array}{l} dx_1\\ dx_2\\ dx_3 \end{array} \right)= \left( \begin{array}{ccc} 1 & 4 & 7\\ 2 & 5 & 8\\ 3 & 6 & 9\\ \end{array} \right) \left( \begin{array}{l} dy_1\\ dy_2\\ dy_3 \end{array} \right) \]
then
\[ \begin{array}{ccl} \omega &=& \left(1dy_1+4dy_2+7dy_3\right)\wedge \left(2dy_1+5dy_2+8dy_3\right)+ 5\left(1dy_1+4dy_2+7dy_3\right)\wedge \left(3dy_1+6dy_2+9dy_3\right) \\ &=&2dy_1\wedge dy_1+5dy_1\wedge dy_2+\cdots+ 5\cdot 7\cdot 6dx_3\wedge dx_2+ 5\cdot 7\cdot 9dx_3\wedge dx_3+\\ &=& -33dy_1\wedge dy_2-66dy_1\wedge dy_3-33dy_2\wedge dy_3 \end{array} \]
Function pullback() function does all this:
options(kform_symbolic_print = "dx") # uses dx etc in print method
pullback(dx^dy+5*dx^dz, matrix(1:9,3,3))## An alternating linear map from V^2 to R with V=R^3:
## -33 dx^dy -66 dx^dz -33 dy^dzHowever, it is slow and I am not 100% sure that there isn’t a much more efficient way to do such a transformation. There are a few tests in tests/testthat. Here I show that transformations may be inverted using matrix inverses:
## An alternating linear map from V^3 to R with V=R^5:
## val
## 2 4 5 = 2Then we will transform according to matrix M and then transform according to the matrix inverse; the functionality works nicely with magrittr pipes:
## An alternating linear map from V^3 to R with V=R^5:
## val
## 3 4 5 = 0
## 1 2 3 = 0
## 2 3 5 = 0
## 2 3 4 = 0
## 2 4 5 = 2
## 1 2 5 = 0
## 1 3 4 = 0
## 1 2 4 = 0
## 1 3 5 = 0
## 1 4 5 = 0Above we see many rows with values small enough for the print method to print an exact zero, but not sufficiently small to be eliminated by the spray internals. We can remove the small entries with zap():
## An alternating linear map from V^3 to R with V=R^5:
## val
## 2 4 5 = 2See how the result is equal to the original \(k\)-form \(2dy_2\wedge dy_4\wedge dy_5\).
Given a \(k\)-form \(\omega\), Spivak defines the differential of \(\omega\) to be a \((k+1)\)-form \(d\omega\) as follows. If
\[ \omega = \sum_{ i_1 < i_2 <\cdots<i_k} \omega_{i_1i_2\ldots i_k} dx^{i_1}\wedge dx^{i_2}\wedge\cdots\wedge dx^{i_k} \]
then
\[ d\omega = \sum_{ i_1 < i_2 <\cdots<i_k} \sum_{\alpha=1}^n D_\alpha\left(\omega_{i_1i_2\ldots i_k}\right) \cdot dx^{i_1}\wedge dx^{i_2}\wedge\cdots\wedge dx^{i_k} \]
where \(D_if(a)=\lim_{h\longrightarrow 0}\frac{f(a^1,\ldots,a^i+h,\ldots,a^n)-f(a^1,\ldots,a^i,\ldots,a^n)}{h}\) is the ordinary \(i^\mathrm{th}\) partial derivative (Spivak, p25). Hubbard and Hubbard take a conceptually distinct approach and define the exterior derivative \(d\phi\) (they use a bold font, \(\mathbf{d}\phi\)) of the \(k\)-form \(\phi\) as the \((k+1)\)-form given by
\[ {d}\phi \left({v}_i,\ldots,{v}_{k+1}\right) = \lim_{h\longrightarrow 0}\frac{1}{h^{k+1}}\int_{\partial P_{x}\left(h{v}_1,\ldots,h{v}_{k+1}\right)}\phi \]
which, by their own account, is a rather opaque mathematical idiom. However, the definition makes sense and it is consistent with Spivak’s definition above. The definition allows one to express the fundamental theorem of calculus in an arbitrary number of dimensions without modification.
It can be shown that
\[ {d}\left(f\,dx_{i_1}\wedge\cdots\wedge dx_{i_k}\right)= {d}f\wedge dx_{i_1}\wedge\cdots\wedge dx_{i_k} \]
where \(f\colon\mathbb{R}^n\longrightarrow\mathbb{R}\) is a scalar function of position. The package provides grad() which, when given a vector \(x_1,\ldots,x_n\) returns the one-form
\[ \sum_{i=1}^n x_idx_i \]
This is useful because \(df=\sum_{j=1}^n\left(D_j f\right)\,dx_j\). Thus
## An alternating linear map from V^1 to R with V=R^4:
## val
## 4 = 1.5
## 3 = -3.2
## 2 = 0.1
## 1 = 0.4We will use the grad() function to verify that, in \(\mathbb{R}^n\), a certain \((k-1)\)-form has zero work function. Motivated by the fact that
\[ F_3=\frac{1}{\left(x^2+y^2+z^2\right)^{3/2}} \begin{pmatrix}x\\y\\z\end{pmatrix} \]
is a divergenceless velocity field in \(\mathbb{R}^3\), H&H go on to define [page 548, equation 6.7.16]
\[ \omega_n= d\frac{1}{\left(x_1^2+\ldots +x_n^2\right)^{n/2}}\sum_{i=1}^{n}(-1)^{i-1} x_idx_1\wedge\cdots\wedge\widehat{dx_i}\wedge\cdots\wedge dx_n \]
(where a hat indicates the absence of a term), and show analytically that \(d\omega=0\). Here I show this using R idiom. The first thing is to define a function that implements the hat:
So, for example:
## An alternating linear map from V^4 to R with V=R^5:
## val
## 1 2 3 4 = 1
## 1 2 3 5 = 1
## 1 2 4 5 = 1
## 1 3 4 5 = 1
## 2 3 4 5 = 1Then we can use the grad() function to calculate \(d\omega\), using the quotient law to express the derivatives analytically:
df <- function(x){
n <- length(x)
S <- sum(x^2)
grad(rep(c(1,-1),length=n)*(S^(n/2) - n*x^2*S^(n/2-1))/S^n
)
}Thus
## An alternating linear map from V^1 to R with V=R^5:
## val
## 5 = -5.67e-05
## 4 = 2.03e-05
## 3 = 8.10e-06
## 2 = -2.84e-05
## 1 = 4.05e-05Now we can use the wedge product of the two parts to show that the exterior derivative is zero:
## An alternating linear map from V^9 to R with V=R^9:
## val
## 1 2 3 4 5 6 7 8 9 = 0We can use the package to verify the celebrated fact that, for any \(k\)-form \(\phi\), \(d\left(d\phi\right)=0\). The first step is to define scalar functions f1(), f2(), f3(), all \(0\)-forms:
f1 <- function(w,x,y,z){x + y^3 + x*y*w*z}
f2 <- function(w,x,y,z){w^2*x*y*z + sin(w) + w+z}
f3 <- function(w,x,y,z){w*x*y*z + sin(x) + cos(w)}Now we need to define elementary \(1\)-forms:
I will demonstrate the theorem by defining a \(2\)-form which is the sum of three elementary two-forms, evaluated at a particular point in \(\mathbb{R}^4\):
We could use slightly slicker R idiom by defining elementary forms e1,e2,e3 and then defining phi to be a linear sum, weighted with \(0\)-forms given by the (scalar) functions f1,f2,f3:
e1 <- dw ^ dx
e2 <- dw ^ dy
e3 <- dy ^ dz
phi <-
(
+f1(1,2,3,4) ^ e1
+f2(1,2,3,4) ^ e2
+f3(1,2,3,4) ^ e3
)
phi## An alternating linear map from V^2 to R with V=R^4:
## val
## 1 3 = 29.84147
## 1 2 = 53.00000
## 3 4 = 25.44960Now to evaluate first derivatives of f1() etc at point \((1,2,3,4)\), using Deriv() from the Deriv package:
So Df1 etc are numeric vectors of length 4, for example:
## w x y z
## 24 13 35 6To calculate dphi, or \(d\phi\), we can use function grad():
## An alternating linear map from V^3 to R with V=R^4:
## val
## 1 3 4 = 30.15853
## 1 2 3 = 23.00000
## 1 2 4 = 6.00000
## 2 3 4 = 11.58385Now work on the differential of the differential. First evaluate the Hessians (4x4 numeric matrices) at the same point:
Hf1 <- matrix(Deriv(f1,nderiv=2)(1,2,3,4),4,4)
Hf2 <- matrix(Deriv(f2,nderiv=2)(1,2,3,4),4,4)
Hf3 <- matrix(Deriv(f3,nderiv=2)(1,2,3,4),4,4)For example
## w x y z
## w 0 12 8 6
## x 12 0 4 3
## y 8 4 18 2
## z 6 3 2 0(note the matrix is symmetric; also note carefully the nonzero diagonal term). But \(dd\phi\) is clearly zero as the Hessians are symmetrical:
ij <- expand.grid(seq_len(nrow(Hf1)),seq_len(ncol(Hf1)))
ddphi <- # should be zero
(
+as.kform(ij,c(Hf1))
+as.kform(ij,c(Hf2))
+as.kform(ij,c(Hf3))
)
ddphi## The zero alternating linear map from V^2 to R with V=R^n:
## empty sparse array with 2 columnsas expected.
In its most general form, Stokes’s theorem states
\[ \int_{\partial X}\phi=\int_Xd\phi \]
where \(X\subset\mathbb{R}^n\) is a compact oriented \((k+1)\)-dimensional manifold with boundary \(\partial X\) and \(\phi\) is a \(k\)-form defined on a neighborhood of \(X\).
We will verify Stokes, following 6.9.5 of Hubbard in which
\[ \phi= \left(x_1-x_2^2+x_3^3-\cdots\pm x_n^n\right) \left( \sum_{i=1}^n dx_1\wedge\cdots\wedge\widehat{dx_i}\wedge\cdots\wedge dx_n \right) \]
(a hat indicates that a term is absent), and we wish to evaluate \(\int_{\partial C_a}\phi\) where \(C_a\) is the cube \(0\leq x_j\leq a, 1\leq j\leq n\). Stokes tells us that this is equal to \(\int_{C_a} d\phi\), which is given by
\[ d\phi = \left( 1+2x_2+\cdots + nx_n^{n-1}\right) dx_1\wedge\cdots\wedge dx_n \]
and so the volume integral is just
\[ \sum_{j=1}^n \int_{x_1=0}^a \int_{x_2=0}^a \cdots \int_{x_i=0}^a jx_j^{j-1} dx_1 dx_2\ldots dx_n= a^{n-1}\left(a+a^2+\cdots+a^n\right). \]
Stokes’s theorem, being trivial, is not amenable to direct numerical verification but the package does allow slick creation of \(\phi\):
phi <- function(x){
n <- length(x)
sum(x^seq_len(n)*rep_len(c(1,-1),n)) * as.kform(t(apply(diag(n)<1,2,which)))
}
phi(1:9)## An alternating linear map from V^8 to R with V=R^9:
## val
## 2 3 4 5 6 7 8 9 = 371423053
## 1 2 3 4 5 7 8 9 = 371423053
## 1 3 4 5 6 7 8 9 = 371423053
## 1 2 3 4 6 7 8 9 = 371423053
## 1 2 3 4 5 6 7 9 = 371423053
## 1 2 4 5 6 7 8 9 = 371423053
## 1 2 3 5 6 7 8 9 = 371423053
## 1 2 3 4 5 6 8 9 = 371423053
## 1 2 3 4 5 6 7 8 = 371423053(recall that phi is a function that maps \(\mathbb{R}^9\) to 8-forms. Here we choose \(\left(1,2,\ldots,9\right)\in\mathbb{R}^9\) and phi(1:9) as shown above is the resulting 8-form. Thus, if we write \(\phi_{1:9}\) for phi(1:9) we would have \(\phi_{1:9}\colon\left(\mathbb{R}^9\right)^8\longrightarrow\mathbb{R}\), with package idiom as follows:
## [1] -26620528Further, \(d\phi\) is given by
## An alternating linear map from V^9 to R with V=R^9:
## val
## 1 2 3 4 5 6 7 8 9 = 405071317(observe that dphi(1:9) is a 9-form, with \(d\phi_{1:9}\colon\left(\mathbb{R}^9\right)^9\longrightarrow\mathbb{R}\)). Now consider Spivak’s theorem 4.6 (page 82), which in this context states that a 9-form is proportional to the determinant of the \(9\times 9\) matrix formed from its arguments, with constant of proportionality equal to the form evaluated on the identity matrix \(I_9\) [formally and more generally, if \(v_1,\ldots,v_n\) is a basis for \(V\), \(\omega\in\Lambda^n(V)\) and \(w_i=\sum a_{ij}v_j\) then \(\omega\left(w_1,\ldots,w_n\right) = \det\left(a_{ij}\right)\cdot\omega\left(v_1,\ldots v_n\right)\)]. Numerically:
## [1] -9850953## [1] -9850953