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DifferentialGeometry

  

ExteriorDerivative

  

take the exterior derivative of a differential form

 

Calling Sequence

Parameters

Description

Examples

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));

ax,y,zwill now be displayed asa

bx,y,zwill now be displayed asb

cx,y,zwill now be displayed asc

(1)

The exterior derivative of a function:

ExteriorDerivative(a(x, y, z));

_DGform,M,1,1,diffax,y,z,x,2,diffax,y,z,y,3,diffax,y,z,z

(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);

_DGform,M,1,1,ax,y,z,2,bx,y,z,3,cx,y,z

(3)

ExteriorDerivative(omega1);

_DGform,M,2,1,2,diffbx,y,z,xdiffax,y,z,y,1,3,diffcx,y,z,xdiffax,y,z,z,2,3,diffcx,y,z,ydiffbx,y,z,z

(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);

_DGform,M,2,1,2,cx,y,z,1,3,bx,y,z,2,3,ax,y,z

(5)

ExteriorDerivative(omega2);

_DGform,M,3,1,2,3,diffax,y,z,x+diffbx,y,z,y+diffcx,y,z,z

(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);

_DGform,M,1,1,expycosz,3,lnxsiny

(7)

omega4 := ExteriorDerivative(omega3);

_DGform,M,2,1,2,expycosz,1,3,expysinzx+sinyx,2,3,lnxcosy

(8)

ExteriorDerivative(omega4);

_DGform,M,3,1,2,3,0

(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]);

_DGform,M,2,1,2,1,_DGform,M,3,1,2,3,1

(10)

A := Matrix([[y &mult dx, z &mult dy], [0 &mult dx, dz]]);

_DGform,M,1,1,y_DGform,M,1,2,z_DGform,M,1,1,0_DGform,M,1,3,1

(11)

ExteriorDerivative(A);

_DGform,M,2,1,2,1_DGform,M,2,2,3,1_DGform,M,2,1,2,0_DGform,M,2,1,2,0

(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]);

_DGform,M,1,1,x,2,y,3,z,_DGform,M,1,2,x,3,y,_DGform,M,1,3,x

(13)

FrData := FrameData(Fr, P);

FrDatadΘ1=0,dΘ2=Θ1Θ2x2yΘ1Θ3x3+x+zΘ2Θ3x3,dΘ3=Θ1Θ3x2yΘ2Θ3x3

(14)

DGsetup(FrData);

frame name: P

(15)

ExteriorDerivative(z);

_DGform,P,1,3,1x

(16)

ExteriorDerivative(x*Theta1 + y^2*Theta2);

_DGform,P,2,1,2,yy+1x2,1,3,y3+xzy2x3,2,3,y23x+zx3

(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);

LDe1,e3=e1,e1,e4=e2,e2,e3=e2,e2,e4=e1

(18)

DGsetup(LD);

Lie algebra: Alg1

(19)

ExteriorDerivative(theta1);

_DGform,Alg1,2,1,3,1,2,4,1

(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);

frame name: M11

(21)

ExteriorDerivative(alpha);

_DGform,M11,2,2,f

(22)

ExteriorDerivative(alpha &w beta &w beta);

_DGform,M11,6,1,2,3,2,2,2,2,f

(23)
M11 > 

 

See Also

DifferentialGeometry

LieBracket

DeRhamHomotopy

PDEtools[declare]

Physics[ExteriorDerivative]

Physics