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DifferentialGeometry

 ExteriorDerivative
 take the exterior derivative of a differential form

 Calling Sequence ExteriorDerivative(omega)

Parameters

 omega - a Maple expression or a differential form

Description

 • The exterior derivative of a differential p-form omega is a differential form d(omega) of degree p + 1.  There are two standard ways to intrinsically define the exterior derivative d.
 • The exterior derivative can be defined directly in terms of the Lie bracket.  For a 1-form alpha and a 2-form beta this definition is:

d(alpha)(X, Y) = X(alpha(Y)) - Y(alpha(X)) - alpha([X, Y]),

d(beta)(X, Y, Z) = X(beta(Y, Z)) - Y(beta(X, Z)) + Z(beta(X, Y)) - X(beta([Y, Z])) + Y(beta([X, Z ])) - Z(beta([X, Y])),

where X, Y, Z are vector fields.  Most of the references listed on the DifferentialGeometry References page contain the general formula for the exterior derivative of a p-form.

 • Alternatively, d can be defined uniquely as that linear operator acting on differential forms such that:

[i]  for functions f, d(f)(X) = X(f),  where X  is any vector field;

[ii]  d(alpha &w beta) = d(alpha) &w beta + (- 1)^p alpha &w d(beta), where alpha and beta are differential forms and p is the degree of alpha; and

[iii] d(d(alpha)) = 0.

The explicit coordinate formulas for the exterior derivatives of a function, a 1-form and a 2-form in 3 dimensions are given in Example 1.

 • The ExteriorDerivative command can also be applied to a list of differential forms.
 • This command is part of the DifferentialGeometry package, and so can be used in the form ExteriorDerivative(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-ExteriorDerivative.

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

We initialize a 3-dimensional manifold with coordinates [x, y, z].

We use the declare command in  PDEtools to display the partial derivatives of the functions a(x, y, z), b(x, y, z) and c(x, y, z) in compact form.

 > DGsetup([x, y, z], M):
 > PDEtools[declare](a(x, y, z), b(x, y, z), c(x, y, z));
 ${a}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{will now be displayed as}}{}{a}$
 ${b}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{will now be displayed as}}{}{b}$
 ${c}{}\left({x}{,}{y}{,}{z}\right){}{\mathrm{will now be displayed as}}{}{c}$ (1)

The exterior derivative of a function:

 > ExteriorDerivative(a(x, y, z));
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{\mathrm{diff}}{}\left({a}{}\left({x}{,}{y}{,}{z}\right){,}{x}\right)\right]{,}\left[\left[{2}\right]{,}{\mathrm{diff}}{}\left({a}{}\left({x}{,}{y}{,}{z}\right){,}{y}\right)\right]{,}\left[\left[{3}\right]{,}{\mathrm{diff}}{}\left({a}{}\left({x}{,}{y}{,}{z}\right){,}{z}\right)\right]\right]\right]\right)$ (2)

The exterior derivative of a 1-form:

 > omega1 := evalDG(a(x, y, z)*dx + b(x, y, z)*dy + c(x, y, z)*dz);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{a}{}\left({x}{,}{y}{,}{z}\right)\right]{,}\left[\left[{2}\right]{,}{b}{}\left({x}{,}{y}{,}{z}\right)\right]{,}\left[\left[{3}\right]{,}{c}{}\left({x}{,}{y}{,}{z}\right)\right]\right]\right]\right)$ (3)
 > ExteriorDerivative(omega1);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{\mathrm{diff}}{}\left({b}{}\left({x}{,}{y}{,}{z}\right){,}{x}\right){-}\left({\mathrm{diff}}{}\left({a}{}\left({x}{,}{y}{,}{z}\right){,}{y}\right)\right)\right]{,}\left[\left[{1}{,}{3}\right]{,}{\mathrm{diff}}{}\left({c}{}\left({x}{,}{y}{,}{z}\right){,}{x}\right){-}\left({\mathrm{diff}}{}\left({a}{}\left({x}{,}{y}{,}{z}\right){,}{z}\right)\right)\right]{,}\left[\left[{2}{,}{3}\right]{,}{\mathrm{diff}}{}\left({c}{}\left({x}{,}{y}{,}{z}\right){,}{y}\right){-}\left({\mathrm{diff}}{}\left({b}{}\left({x}{,}{y}{,}{z}\right){,}{z}\right)\right)\right]\right]\right]\right)$ (4)

The exterior derivative of a 2-form:

 > omega2 :=  evalDG(c(x, y, z)*dx &w dy - b(x, y, z)*dx &w dz + a(x, y, z)*dy &w dz);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{c}{}\left({x}{,}{y}{,}{z}\right)\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}{b}{}\left({x}{,}{y}{,}{z}\right)\right]{,}\left[\left[{2}{,}{3}\right]{,}{a}{}\left({x}{,}{y}{,}{z}\right)\right]\right]\right]\right)$ (5)
 > ExteriorDerivative(omega2);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{3}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{\mathrm{diff}}{}\left({a}{}\left({x}{,}{y}{,}{z}\right){,}{x}\right){+}{\mathrm{diff}}{}\left({b}{}\left({x}{,}{y}{,}{z}\right){,}{y}\right){+}{\mathrm{diff}}{}\left({c}{}\left({x}{,}{y}{,}{z}\right){,}{z}\right)\right]\right]\right]\right)$ (6)

Example 2.

By way of an example, we illustrate the fact that d^2 = 0.

 > omega3 := evalDG(exp(y)*cos(z)*dx + ln(x)*sin(y)*dz);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{\mathrm{exp}}{}\left({y}\right){}{\mathrm{cos}}{}\left({z}\right)\right]{,}\left[\left[{3}\right]{,}{\mathrm{ln}}{}\left({x}\right){}{\mathrm{sin}}{}\left({y}\right)\right]\right]\right]\right)$ (7)
 > omega4 := ExteriorDerivative(omega3);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{\mathrm{exp}}{}\left({y}\right){}{\mathrm{cos}}{}\left({z}\right)\right]{,}\left[\left[{1}{,}{3}\right]{,}\frac{{\mathrm{exp}}{}\left({y}\right){}{\mathrm{sin}}{}\left({z}\right){}{x}{+}{\mathrm{sin}}{}\left({y}\right)}{{x}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{\mathrm{ln}}{}\left({x}\right){}{\mathrm{cos}}{}\left({y}\right)\right]\right]\right]\right)$ (8)
 > ExteriorDerivative(omega4);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{3}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{0}\right]\right]\right]\right)$ (9)

Example 3.

The ExteriorDerivative command can also be applied to a list of forms or a matrix of forms.

 > ExteriorDerivative([y*dx, z*dx &w dy]);
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{3}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (10)
 > A := Matrix([[y &mult dx, z &mult dy], [0 &mult dx, dz]]);
 $\left[\begin{array}{cc}\mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],y\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[2\right],z\right]\right]\right]\right)\\ \mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],0\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[3\right],1\right]\right]\right]\right)\end{array}\right]$ (11)
 > ExteriorDerivative(A);
 $\left[\begin{array}{cc}\mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[1,2\right],-1\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[2,3\right],-1\right]\right]\right]\right)\\ \mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[1,2\right],0\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[1,2\right],0\right]\right]\right]\right)\end{array}\right]$ (12)

Example 4.

The ExteriorDerivative command can also be used with adapted frames.  First we define an adapted coframe for M.

 > Fr:= evalDG([x*dx + y*dy + z*dz, x*dy + y*dz, x*dz]);
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{x}\right]{,}\left[\left[{2}\right]{,}{y}\right]{,}\left[\left[{3}\right]{,}{z}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{x}\right]{,}\left[\left[{3}\right]{,}{y}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{x}\right]\right]\right]\right)\right]$ (13)
 > FrData := FrameData(Fr, P);
 ${\mathrm{FrData}}{≔}\left[{d}{}{\mathrm{Θ1}}{=}{0}{,}{d}{}{\mathrm{Θ2}}{=}\frac{{\mathrm{Θ1}}{}{\bigwedge }{}{\mathrm{Θ2}}}{{{x}}^{{2}}}{-}\frac{{y}{}{\mathrm{Θ1}}{}{\bigwedge }{}{\mathrm{Θ3}}}{{{x}}^{{3}}}{+}\frac{\left({x}{+}{z}\right){}{\mathrm{Θ2}}{}{\bigwedge }{}{\mathrm{Θ3}}}{{{x}}^{{3}}}{,}{d}{}{\mathrm{Θ3}}{=}\frac{{\mathrm{Θ1}}{}{\bigwedge }{}{\mathrm{Θ3}}}{{{x}}^{{2}}}{-}\frac{{y}{}{\mathrm{Θ2}}{}{\bigwedge }{}{\mathrm{Θ3}}}{{{x}}^{{3}}}\right]$ (14)
 > DGsetup(FrData);
 ${\mathrm{frame name: P}}$ (15)
 > ExteriorDerivative(z);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}\frac{{1}}{{x}}\right]\right]\right]\right)$ (16)
 > ExteriorDerivative(x*Theta1 + y^2*Theta2);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}\frac{{y}{}\left({y}{+}{1}\right)}{{{x}}^{{2}}}\right]{,}\left[\left[{1}{,}{3}\right]{,}\frac{{-}{{y}}^{{3}}{+}{x}{}{z}{-}{{y}}^{{2}}}{{{x}}^{{3}}}\right]{,}\left[\left[{2}{,}{3}\right]{,}\frac{{{y}}^{{2}}{}\left({3}{}{x}{+}{z}\right)}{{{x}}^{{3}}}\right]\right]\right]\right)$ (17)

Example 5.

The ExteriorDerivative command can be used with Lie algebras.

 > LD := LieAlgebraData([[x1, x3] = x1, [x1, x4] = -x2, [x2, x3] = x2, [x2, x4] = x1], [x1, x2, x3, x4], Alg1);
 ${\mathrm{LD}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}\right]$ (18)
 > DGsetup(LD);
 ${\mathrm{Lie algebra: Alg1}}$ (19)
 > ExteriorDerivative(theta1);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{4}\right]{,}{-}{1}\right]\right]\right]\right)$ (20)

Example 6.

The ExteriorDerivative command can also be used with abstract differential forms.

 > DGsetup([f = dgform(0), alpha = dgform(1), beta = dgform(2)], [d(alpha) = f*beta], M11);
 ${\mathrm{frame name: M11}}$ (21)
 > ExteriorDerivative(alpha);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{M11}}{,}{2}\right]{,}\left[\left[\left[{2}\right]{,}{f}\right]\right]\right]\right)$ (22)
 > ExteriorDerivative(alpha &w beta &w beta);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{M11}}{,}{6}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{-}{2}\right]{,}\left[\left[{2}{,}{2}{,}{2}\right]{,}{f}\right]\right]\right]\right)$ (23)
 M11 >