Notebook

SageMath para físicos¶

Rogério T. C.

Aula IV.1¶

Sagemanifolds, Minkowski e Maxwell¶

In [1]:

reset()
%display latex

Espaço de Minkoski e dualidade¶

In [2]:

M.<t,x,y,z> = manifolds.Minkowski()

In [3]:

print(M)

4-dimensional Lorentzian manifold M

In [4]:

print(M.frames())
M.frames()

[Coordinate frame (M, (d/dt,d/dx,d/dy,d/dz))]

Out[4]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(M, \left(\frac{\partial}{\partial t },\frac{\partial}{\partial x },\frac{\partial}{\partial y },\frac{\partial}{\partial z }\right)\right)\right]$

In [5]:

print(M.coframes())
M.coframes()

[Coordinate coframe (M, (dt,dx,dy,dz))]

Out[5]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left(M, \left(\mathrm{d} t,\mathrm{d} x,\mathrm{d} y,\mathrm{d} z\right)\right)\right]$

Calculando

$dt\left(\frac{\partial}{\partial x}\right)$

In [6]:

M.coframes()[0][1](M.frames()[0][1]).display()

Out[6]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} \mathrm{d} x \left( \frac{\partial}{\partial x } \right) : & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, y, z\right) & \longmapsto & 1 \end{array}$

In [7]:

for i in range(2):
    for j in range(2):
        show(M.coframes()[0][i](M.frames()[0][j]).display())

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} \mathrm{d} t \left( \frac{\partial}{\partial t } \right) : & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, y, z\right) & \longmapsto & 1 \end{array}$

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} \mathrm{d} t \left( \frac{\partial}{\partial x } \right) : & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, y, z\right) & \longmapsto & 0 \end{array}$

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} \mathrm{d} x \left( \frac{\partial}{\partial t } \right) : & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, y, z\right) & \longmapsto & 0 \end{array}$

In [8]:

eta = M.metric()
eta.set_name('eta', latex_name=r'\eta')

In [9]:

eta.display()

Out[9]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\eta = -\mathrm{d} t\otimes \mathrm{d} t+\mathrm{d} x\otimes \mathrm{d} x+\mathrm{d} y\otimes \mathrm{d} y+\mathrm{d} z\otimes \mathrm{d} z$

In [10]:

eta[:]

Out[10]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

$dx^i\left(\frac{\partial}{\partial x^j}\right) = \eta\left(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right)$

In [11]:

M.coframes()[0][1](M.frames()[0][1])==eta(M.frames()[0][1],M.frames()[0][1])

Out[11]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{True}$

$dt\left(\frac{\partial}{\partial t}\right) \neq \eta\left(\frac{\partial}{\partial t},\frac{\partial}{\partial t}\right)$

In [12]:

M.frames()[0][0].down(eta).display()

Out[12]:

$\newcommand{\Bold}[1]{\mathbf{#1}}-\mathrm{d} t$

In [13]:

M.coframes()[0][0](M.frames()[0][0])==eta(M.frames()[0][0],M.frames()[0][0])

Out[13]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{False}$

Formas diferencias e equações de Maxwell¶

$E$ , $B$ estáticos¶

In [14]:

vE0 = [function('E_'+i)(x,y,z) for i in ['x','y','z']];vE0

Out[14]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[E_{x}\left(x, y, z\right), E_{y}\left(x, y, z\right), E_{z}\left(x, y, z\right)\right]$

In [15]:

E0 = M.one_form([0]+vE0, name='E');E0.display()

Out[15]:

$\newcommand{\Bold}[1]{\mathbf{#1}}E = E_{x}\left(x, y, z\right) \mathrm{d} x + E_{y}\left(x, y, z\right) \mathrm{d} y + E_{z}\left(x, y, z\right) \mathrm{d} z$

In [16]:

dE0 = E0.exterior_derivative();dE0.display()

Out[16]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{d}E = \left( -\frac{\partial\,E_{x}}{\partial y} + \frac{\partial\,E_{y}}{\partial x} \right) \mathrm{d} x\wedge \mathrm{d} y + \left( -\frac{\partial\,E_{x}}{\partial z} + \frac{\partial\,E_{z}}{\partial x} \right) \mathrm{d} x\wedge \mathrm{d} z + \left( -\frac{\partial\,E_{y}}{\partial z} + \frac{\partial\,E_{z}}{\partial y} \right) \mathrm{d} y\wedge \mathrm{d} z$

In [17]:

vB0 = [function('B_z')(x,y,z), -function('B_y')(x,y,z), function('B_x')(x,y,z)];vB0

Out[17]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[B_{z}\left(x, y, z\right), -B_{y}\left(x, y, z\right), B_{x}\left(x, y, z\right)\right]$

In [18]:

B0 = M.diff_form(2,name='B')
B0[1,2],B0[1,3],B0[2,3] = vB0

In [19]:

B0.display() 

Out[19]:

$\newcommand{\Bold}[1]{\mathbf{#1}}B = B_{z}\left(x, y, z\right) \mathrm{d} x\wedge \mathrm{d} y -B_{y}\left(x, y, z\right) \mathrm{d} x\wedge \mathrm{d} z + B_{x}\left(x, y, z\right) \mathrm{d} y\wedge \mathrm{d} z$

In [20]:

dB0 = B0.exterior_derivative();dB0.display()

Out[20]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{d}B = \left( \frac{\partial\,B_{x}}{\partial x} + \frac{\partial\,B_{y}}{\partial y} + \frac{\partial\,B_{z}}{\partial z} \right) \mathrm{d} x\wedge \mathrm{d} y\wedge \mathrm{d} z$

Equaões de Maxwell no Vácuo (caso estático)

Lei	Equação diferencial
Lei de Gauss	$\nabla \cdot \mathbf{E} =0$
Lei de Gauss para o magnetismo	$\nabla \cdot \mathbf{B} =0$
Lei de Faraday	$\nabla \times \mathbf{E} = 0$
Lei circular de Ampère	$\nabla \times \mathbf{B} =0$

Lei de Faraday: $\quad d\mathbf{E} = 0$

In [21]:

dE0.display()

Out[21]:

Lei de Gauss: $\quad d\mathbf{B} = 0$

In [22]:

dB0.display()

Out[22]:

Caso dinâmico¶

In [23]:

vE = [function('E_'+i)(t,x,y,z) for i in ['x','y','z']]

In [24]:

vB = [function('B_z')(t,x,y,z), -function('B_y')(t,x,y,z), function('B_x')(t,x,y,z)];vB0

Out[24]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[B_{z}\left(x, y, z\right), -B_{y}\left(x, y, z\right), B_{x}\left(x, y, z\right)\right]$

In [25]:

E = M.one_form([0]+vE, name='E');E.display()

Out[25]:

$\newcommand{\Bold}[1]{\mathbf{#1}}E = E_{x}\left(t, x, y, z\right) \mathrm{d} x + E_{y}\left(t, x, y, z\right) \mathrm{d} y + E_{z}\left(t, x, y, z\right) \mathrm{d} z$

In [26]:

B = M.diff_form(2,name='B')
B[1,2],B[1,3],B[2,3] = vB;B.display()

Out[26]:

$\newcommand{\Bold}[1]{\mathbf{#1}}B = B_{z}\left(t, x, y, z\right) \mathrm{d} x\wedge \mathrm{d} y -B_{y}\left(t, x, y, z\right) \mathrm{d} x\wedge \mathrm{d} z + B_{x}\left(t, x, y, z\right) \mathrm{d} y\wedge \mathrm{d} z$

In [27]:

dt = M.coframes()[0][0];dt

Out[27]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{d} t$

In [28]:

F = E.wedge(dt) + B;F.display()

Out[28]:

$\newcommand{\Bold}[1]{\mathbf{#1}}E\wedge \mathrm{d} t + B = -E_{x}\left(t, x, y, z\right) \mathrm{d} t\wedge \mathrm{d} x -E_{y}\left(t, x, y, z\right) \mathrm{d} t\wedge \mathrm{d} y -E_{z}\left(t, x, y, z\right) \mathrm{d} t\wedge \mathrm{d} z + B_{z}\left(t, x, y, z\right) \mathrm{d} x\wedge \mathrm{d} y -B_{y}\left(t, x, y, z\right) \mathrm{d} x\wedge \mathrm{d} z + B_{x}\left(t, x, y, z\right) \mathrm{d} y\wedge \mathrm{d} z$

In [29]:

F[:]

Out[29]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 0 & -E_{x}\left(t, x, y, z\right) & -E_{y}\left(t, x, y, z\right) & -E_{z}\left(t, x, y, z\right) \\ E_{x}\left(t, x, y, z\right) & 0 & B_{z}\left(t, x, y, z\right) & -B_{y}\left(t, x, y, z\right) \\ E_{y}\left(t, x, y, z\right) & -B_{z}\left(t, x, y, z\right) & 0 & B_{x}\left(t, x, y, z\right) \\ E_{z}\left(t, x, y, z\right) & B_{y}\left(t, x, y, z\right) & -B_{x}\left(t, x, y, z\right) & 0 \end{array}\right)$

In [30]:

dF = F.exterior_derivative();dF.display()

Out[30]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{d} \left( E\wedge \mathrm{d} t + B \right) = \left( \frac{\partial\,B_{z}}{\partial t} - \frac{\partial\,E_{x}}{\partial y} + \frac{\partial\,E_{y}}{\partial x} \right) \mathrm{d} t\wedge \mathrm{d} x\wedge \mathrm{d} y + \left( -\frac{\partial\,B_{y}}{\partial t} - \frac{\partial\,E_{x}}{\partial z} + \frac{\partial\,E_{z}}{\partial x} \right) \mathrm{d} t\wedge \mathrm{d} x\wedge \mathrm{d} z + \left( \frac{\partial\,B_{x}}{\partial t} - \frac{\partial\,E_{y}}{\partial z} + \frac{\partial\,E_{z}}{\partial y} \right) \mathrm{d} t\wedge \mathrm{d} y\wedge \mathrm{d} z + \left( \frac{\partial\,B_{x}}{\partial x} + \frac{\partial\,B_{y}}{\partial y} + \frac{\partial\,B_{z}}{\partial z} \right) \mathrm{d} x\wedge \mathrm{d} y\wedge \mathrm{d} z$

Lei	Equação diferencial
Lei de Gauss para o magnetismo	$\nabla \cdot \mathbf{B} =0$
Lei de Faraday	$\nabla \times \mathbf{E} =-{\frac {\partial \mathbf {B} }{\partial t}}$

Lei de Gauss para $\mathbf{B}$ e Lei de Faraday: $\quad dF =0$

Dual de Hodge

$\star: \Omega^p(M) \to \Omega^{n-p}$ tal que

$\omega \wedge \star \mu = g(\omega,\mu) \text{vol}$

In [31]:

star_F = F.hodge_dual(eta)

In [32]:

star_F.display()

Out[32]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\star \left( E\wedge \mathrm{d} t + B \right) = B_{x}\left(t, x, y, z\right) \mathrm{d} t\wedge \mathrm{d} x + B_{y}\left(t, x, y, z\right) \mathrm{d} t\wedge \mathrm{d} y + B_{z}\left(t, x, y, z\right) \mathrm{d} t\wedge \mathrm{d} z + E_{z}\left(t, x, y, z\right) \mathrm{d} x\wedge \mathrm{d} y -E_{y}\left(t, x, y, z\right) \mathrm{d} x\wedge \mathrm{d} z + E_{x}\left(t, x, y, z\right) \mathrm{d} y\wedge \mathrm{d} z$

In [33]:

dstar_F = star_F.exterior_derivative().hodge_dual(eta)

In [34]:

dstar_F.display()

Out[34]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\star \mathrm{d} \star \left( E\wedge \mathrm{d} t + B \right) = \left( -\frac{\partial\,E_{x}}{\partial x} - \frac{\partial\,E_{y}}{\partial y} - \frac{\partial\,E_{z}}{\partial z} \right) \mathrm{d} t + \left( -\frac{\partial\,B_{y}}{\partial z} + \frac{\partial\,B_{z}}{\partial y} - \frac{\partial\,E_{x}}{\partial t} \right) \mathrm{d} x + \left( \frac{\partial\,B_{x}}{\partial z} - \frac{\partial\,B_{z}}{\partial x} - \frac{\partial\,E_{y}}{\partial t} \right) \mathrm{d} y + \left( -\frac{\partial\,B_{x}}{\partial y} + \frac{\partial\,B_{y}}{\partial x} - \frac{\partial\,E_{z}}{\partial t} \right) \mathrm{d} z$

Fonte

In [35]:

ep0 = var('epsilon_0')
mu0 = var('mu_0')

In [36]:

mu0

Out[36]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\mu_{0}$

In [37]:

rho = function('rho', latex_name=r'\rho')(t,x,y,z)
fJ = [rho]+[function('J_'+str(i))(t,x,y,z) for i in range(1,4)]

In [38]:

J = M.one_form(fJ, name='J')

In [39]:

J.display()

Out[39]:

$\newcommand{\Bold}[1]{\mathbf{#1}}J = \rho\left(t, x, y, z\right) \mathrm{d} t + J_{1}\left(t, x, y, z\right) \mathrm{d} x + J_{2}\left(t, x, y, z\right) \mathrm{d} y + J_{3}\left(t, x, y, z\right) \mathrm{d} z$

In [40]:

(dstar_F+J).display()

Out[40]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\star \mathrm{d} \star \left( E\wedge \mathrm{d} t + B \right) + J = \left( \rho\left(t, x, y, z\right) - \frac{\partial\,E_{x}}{\partial x} - \frac{\partial\,E_{y}}{\partial y} - \frac{\partial\,E_{z}}{\partial z} \right) \mathrm{d} t + \left( J_{1}\left(t, x, y, z\right) - \frac{\partial\,B_{y}}{\partial z} + \frac{\partial\,B_{z}}{\partial y} - \frac{\partial\,E_{x}}{\partial t} \right) \mathrm{d} x + \left( J_{2}\left(t, x, y, z\right) + \frac{\partial\,B_{x}}{\partial z} - \frac{\partial\,B_{z}}{\partial x} - \frac{\partial\,E_{y}}{\partial t} \right) \mathrm{d} y + \left( J_{3}\left(t, x, y, z\right) - \frac{\partial\,B_{x}}{\partial y} + \frac{\partial\,B_{y}}{\partial x} - \frac{\partial\,E_{z}}{\partial t} \right) \mathrm{d} z$

Lei	Equação diferencial
Lei de Gauss	$\nabla \cdot \mathbf{E} ={\rho }$
Lei de Gauss para o magnetismo	$\nabla \cdot \mathbf{B} =0$
Lei de Faraday	$\nabla \times \mathbf{E} =-{\frac {\partial \mathbf {B} }{\partial t}}$
Lei circular de Ampère	$\nabla \times \mathbf{B} =\mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}$

Lei de Gauss e Lei de Ampère

$\star d \star dF = \mathbf{J}$

Lei de conservação¶

In [41]:

J.hodge_dual(eta).exterior_derivative().display()

Out[41]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{d}\star J = \left( \frac{\partial\,J_{1}}{\partial x} + \frac{\partial\,J_{2}}{\partial y} + \frac{\partial\,J_{3}}{\partial z} - \frac{\partial\,\rho}{\partial t} \right) \mathrm{d} t\wedge \mathrm{d} x\wedge \mathrm{d} y\wedge \mathrm{d} z$

Potencial¶

In [42]:

Phi = function('Phi', latex_name=r'\Phi')(t,x,y,z)

In [43]:

fA = [-Phi]+[function('A_'+str(i))(t,x,y,z) for i in range(1,4)]

In [44]:

fA

Out[44]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[-\Phi\left(t, x, y, z\right), A_{1}\left(t, x, y, z\right), A_{2}\left(t, x, y, z\right), A_{3}\left(t, x, y, z\right)\right]$

In [45]:

A = M.one_form(fA, name='A')

In [46]:

A.display()

Out[46]:

$\newcommand{\Bold}[1]{\mathbf{#1}}A = -\Phi\left(t, x, y, z\right) \mathrm{d} t + A_{1}\left(t, x, y, z\right) \mathrm{d} x + A_{2}\left(t, x, y, z\right) \mathrm{d} y + A_{3}\left(t, x, y, z\right) \mathrm{d} z$

In [47]:

print(A)
A.parent()

1-form A on the 4-dimensional Lorentzian manifold M

Out[47]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\Omega^{1}\left(M\right)$

In [48]:

F_A = A.exterior_derivative()

In [49]:

F_A.display()

Out[49]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\mathrm{d}A = \left( \frac{\partial\,A_{1}}{\partial t} + \frac{\partial\,\Phi}{\partial x} \right) \mathrm{d} t\wedge \mathrm{d} x + \left( \frac{\partial\,A_{2}}{\partial t} + \frac{\partial\,\Phi}{\partial y} \right) \mathrm{d} t\wedge \mathrm{d} y + \left( \frac{\partial\,A_{3}}{\partial t} + \frac{\partial\,\Phi}{\partial z} \right) \mathrm{d} t\wedge \mathrm{d} z + \left( -\frac{\partial\,A_{1}}{\partial y} + \frac{\partial\,A_{2}}{\partial x} \right) \mathrm{d} x\wedge \mathrm{d} y + \left( -\frac{\partial\,A_{1}}{\partial z} + \frac{\partial\,A_{3}}{\partial x} \right) \mathrm{d} x\wedge \mathrm{d} z + \left( -\frac{\partial\,A_{2}}{\partial z} + \frac{\partial\,A_{3}}{\partial y} \right) \mathrm{d} y\wedge \mathrm{d} z$

In [50]:

F_A[0,1]

Out[50]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\partial\,A_{1}}{\partial t} + \frac{\partial\,\Phi}{\partial x}$

In [51]:

F_A[:]

Out[51]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 0 & \frac{\partial\,A_{1}}{\partial t} + \frac{\partial\,\Phi}{\partial x} & \frac{\partial\,A_{2}}{\partial t} + \frac{\partial\,\Phi}{\partial y} & \frac{\partial\,A_{3}}{\partial t} + \frac{\partial\,\Phi}{\partial z} \\ -\frac{\partial\,A_{1}}{\partial t} - \frac{\partial\,\Phi}{\partial x} & 0 & -\frac{\partial\,A_{1}}{\partial y} + \frac{\partial\,A_{2}}{\partial x} & -\frac{\partial\,A_{1}}{\partial z} + \frac{\partial\,A_{3}}{\partial x} \\ -\frac{\partial\,A_{2}}{\partial t} - \frac{\partial\,\Phi}{\partial y} & \frac{\partial\,A_{1}}{\partial y} - \frac{\partial\,A_{2}}{\partial x} & 0 & -\frac{\partial\,A_{2}}{\partial z} + \frac{\partial\,A_{3}}{\partial y} \\ -\frac{\partial\,A_{3}}{\partial t} - \frac{\partial\,\Phi}{\partial z} & \frac{\partial\,A_{1}}{\partial z} - \frac{\partial\,A_{3}}{\partial x} & \frac{\partial\,A_{2}}{\partial z} - \frac{\partial\,A_{3}}{\partial y} & 0 \end{array}\right)$

In [52]:

print(F_A)
F_A.parent()

2-form dA on the 4-dimensional Lorentzian manifold M

Out[52]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\Omega^{2}\left(M\right)$

Campos de Gauge¶

In [53]:

G = M.scalar_field(function('G')(t,x,y,z))

In [54]:

G.display()

Out[54]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, y, z\right) & \longmapsto & G\left(t, x, y, z\right) \end{array}$

In [55]:

G.exterior_derivative().exterior_derivative().display()

Out[55]:

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

In [56]:

A2 = A+ G.exterior_derivative()
A2.display()

Out[56]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left( -\Phi\left(t, x, y, z\right) + \frac{\partial\,G}{\partial t} \right) \mathrm{d} t + \left( A_{1}\left(t, x, y, z\right) + \frac{\partial\,G}{\partial x} \right) \mathrm{d} x + \left( A_{2}\left(t, x, y, z\right) + \frac{\partial\,G}{\partial y} \right) \mathrm{d} y + \left( A_{3}\left(t, x, y, z\right) + \frac{\partial\,G}{\partial z} \right) \mathrm{d} z$

In [57]:

A2.exterior_derivative().display()

Out[57]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left( \frac{\partial\,A_{1}}{\partial t} + \frac{\partial\,\Phi}{\partial x} \right) \mathrm{d} t\wedge \mathrm{d} x + \left( \frac{\partial\,A_{2}}{\partial t} + \frac{\partial\,\Phi}{\partial y} \right) \mathrm{d} t\wedge \mathrm{d} y + \left( \frac{\partial\,A_{3}}{\partial t} + \frac{\partial\,\Phi}{\partial z} \right) \mathrm{d} t\wedge \mathrm{d} z + \left( -\frac{\partial\,A_{1}}{\partial y} + \frac{\partial\,A_{2}}{\partial x} \right) \mathrm{d} x\wedge \mathrm{d} y + \left( -\frac{\partial\,A_{1}}{\partial z} + \frac{\partial\,A_{3}}{\partial x} \right) \mathrm{d} x\wedge \mathrm{d} z + \left( -\frac{\partial\,A_{2}}{\partial z} + \frac{\partial\,A_{3}}{\partial y} \right) \mathrm{d} y\wedge \mathrm{d} z$

Comparando com a derivada covariante¶

In [58]:

star_dE_star = -(E.hodge_dual(eta).exterior_derivative().hodge_dual(eta))

In [59]:

print(star_dE_star)

Scalar field -*d*E on the 4-dimensional Lorentzian manifold M

In [60]:

star_dE_star.display()

Out[60]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} -\star \mathrm{d}\star E:& M & \longrightarrow & \mathbb{R} \\ & \left(t, x, y, z\right) & \longmapsto & \frac{\partial\,E_{x}}{\partial x} + \frac{\partial\,E_{y}}{\partial y} + \frac{\partial\,E_{z}}{\partial z} \end{array}$

In [61]:

nabla = eta.connection()

In [62]:

nabla(E).display()

Out[62]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\nabla_{\eta} E = \frac{\partial\,E_{x}}{\partial t} \mathrm{d} x\otimes \mathrm{d} t + \frac{\partial\,E_{x}}{\partial x} \mathrm{d} x\otimes \mathrm{d} x + \frac{\partial\,E_{x}}{\partial y} \mathrm{d} x\otimes \mathrm{d} y + \frac{\partial\,E_{x}}{\partial z} \mathrm{d} x\otimes \mathrm{d} z + \frac{\partial\,E_{y}}{\partial t} \mathrm{d} y\otimes \mathrm{d} t + \frac{\partial\,E_{y}}{\partial x} \mathrm{d} y\otimes \mathrm{d} x + \frac{\partial\,E_{y}}{\partial y} \mathrm{d} y\otimes \mathrm{d} y + \frac{\partial\,E_{y}}{\partial z} \mathrm{d} y\otimes \mathrm{d} z + \frac{\partial\,E_{z}}{\partial t} \mathrm{d} z\otimes \mathrm{d} t + \frac{\partial\,E_{z}}{\partial x} \mathrm{d} z\otimes \mathrm{d} x + \frac{\partial\,E_{z}}{\partial y} \mathrm{d} z\otimes \mathrm{d} y + \frac{\partial\,E_{z}}{\partial z} \mathrm{d} z\otimes \mathrm{d} z$

In [63]:

div_E = nabla(E).up(eta,0).trace()

In [64]:

div_E.display()

Out[64]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\begin{array}{llcl} & M & \longrightarrow & \mathbb{R} \\ & \left(t, x, y, z\right) & \longmapsto & \frac{\partial\,E_{x}}{\partial x} + \frac{\partial\,E_{y}}{\partial y} + \frac{\partial\,E_{z}}{\partial z} \end{array}$

In [65]:

star_dE_star == div_E

Out[65]: