# Vector calculus with SageMath¶

## 2. Using spherical coordinates¶

This notebook illustrates operations on vector fields on Euclidean spaces, as introduced in Trac ticket #24623.

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version()

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'SageMath version 8.2.rc4, Release Date: 2018-04-20'
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%display latex

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from sage.manifolds.operators import *   # to get the operators grad, div, curl, etc.


## The 3-dimensional Euclidean space¶

We start by declaring the 3-dimensional Euclidean space $\mathbb{E}^3$, with $(r,\theta,\phi)$ as spherical coordinates:

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E.<r,th,ph> = EuclideanSpace(coordinates='spherical')
print(E)
E

Euclidean space E^3

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$\mathbb{E}^3$ is endowed with the chart of spherical coordinates:

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E.atlas()

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as well as with the associated orthonormal vector frame $(e_r, e_\theta, e_\phi)$:

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E.frames()

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In the above output, $\left(\frac{\partial}{\partial r}, \frac{\partial}{\partial\theta}, \frac{\partial}{\partial \phi}\right)$ is the coordinate frame associated with $(r,\theta,\phi)$; it is not an orthonormal frame and will not be used below.

## Vector fields¶

We define a vector field on $\mathbb{E}^3$ from its components in the orthonormal vector frame $(e_r,e_\theta,e_\phi)$:

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v = E.vector_field(r*sin(2*ph)*sin(th)^2 + r,
r*sin(2*ph)*sin(th)*cos(th),
2*r*cos(ph)^2*sin(th), name='v')
v.display()

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We can access to the components of $v$ via the square bracket operator:

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v

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v[:]

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A vector field can evaluated at any point of $\mathbb{E}^3$:

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p = E((1, pi/2, pi), name='p')
print(p)

Point p on the Euclidean space E^3

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p.coordinates()

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vp = v.at(p)
print(vp)

Vector v at Point p on the Euclidean space E^3

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vp.display()

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We may define a vector field with generic components:

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u = E.vector_field(function('u_r')(r,th,ph),
function('u_theta')(r,th,ph),
function('u_phi')(r,th,ph),
name='u')
u.display()

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u[:]

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up = u.at(p)
up.display()

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## Algebraic operations on vector fields¶

### Dot product¶

The dot (or scalar) product of the vector fields $u$ and $v$ is obtained by the method dot_product, which admits dot as a shortcut alias:

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s = u.dot(v)
s

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In :
print(s)

Scalar field u.v on the Euclidean space E^3


$s= u\cdot v$ is a scalar field, i.e. a map $\mathbb{E}^3 \rightarrow \mathbb{R}$:

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s.display()

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It maps point of $\mathbb{E}^3$ to real numbers:

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s(p)

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Its coordinate expression is

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s.expr()

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### Norm¶

The norm of a vector field is

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s = norm(u)
s

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s.display()

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s.expr()

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The norm is related to the dot product by $\|u\|^2 = u\cdot u$, as we can check:

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norm(u)^2 == u.dot(u)

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For $v$, we have:

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norm(v).expr()

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### Cross product¶

The cross product of $u$ by $v$ is obtained by the method cross_product, which admits cross as a shortcut alias:

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s = u.cross(v)
print(s)

Vector field u x v on the Euclidean space E^3

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s.display()

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### Scalar triple product¶

Let us introduce a third vector field. As a example, we do not pass the components as arguments of vector_field, as we did for $u$ and $v$; instead, we set them in a second stage, via the square bracket operator, any unset component being assumed to be zero:

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w = E.vector_field(name='w')
w = r
w.display()

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The scalar triple product of the vector fields $u$, $v$ and $w$ is obtained as follows:

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triple_product = E.scalar_triple_product()
s = triple_product(u, v, w)
print(s)

Scalar field epsilon(u,v,w) on the Euclidean space E^3

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s.expr()

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Let us check that the scalar triple product of $u$, $v$ and $w$ is $u\cdot(v\times w)$:

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s == u.dot(v.cross(w))

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## Differential operators¶

### Gradient of a scalar field¶

We first introduce a scalar field, via its expression in terms of Cartesian coordinates; in this example, we consider a unspecified function of $(r,\theta,\phi)$:

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F = E.scalar_field(function('f')(r,th,ph), name='F')
F.display()

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The value of $F$ at a point:

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F(p)

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The gradient of $F$:

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print(grad(F))

Vector field grad(F) on the Euclidean space E^3

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grad(F).display()

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norm(grad(F)).display()

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### Divergence¶

The divergence of a vector field:

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s = div(u)
s.display()

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s.expr().expand()

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For $v$ and $w$, we have

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div(v).expr()

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div(w).expr()

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An identity valid for any scalar field $F$ and any vector field $u$:

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div(F*u) == F*div(u) + u.dot(grad(F))

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### Curl¶

The curl of a vector field:

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s = curl(u)
print(s)

Vector field curl(u) on the Euclidean space E^3

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s.display()

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To use the notation rot instead of curl, simply do

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rot = curl


An alternative is

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from sage.manifolds.operators import curl as rot


We have then

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rot(u).display()

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rot(u) == curl(u)

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For $v$ and $w$, we have:

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curl(v).display()

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curl(w).display()

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The curl of a gradient is always zero:

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curl(grad(F)).display()

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The divergence of a curl is always zero:

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div(curl(u)).display()

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An identity valid for any scalar field $F$ and any vector field $u$:

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curl(F*u) == grad(F).cross(u) + F*curl(u)

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### Laplacian¶

The Laplacian of a scalar field:

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s = laplacian(F)
s.display()

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s.expr().expand()

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For a scalar field, the Laplacian is nothing but the divergence of the gradient:

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laplacian(F) == div(grad(F))

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The Laplacian of a vector field:

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Du = laplacian(u)
Du.display()

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Since this expression is quite lengthy, we may ask for a display component by component:

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Du.display_comp()

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We may expand each component:

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for i in E.irange():
Du[i].expand()
Du.display_comp()

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Du

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Du

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Du

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As a test, we may check that these formulas coincide with those of Wikipedia's article Del in cylindrical and spherical coordinates.

For $v$ and $w$, we have

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laplacian(v).display()

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laplacian(w).display()

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We have:

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curl(curl(u)).display()

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grad(div(u)).display()

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and we may check a famous identity:

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curl(curl(u)) == grad(div(u)) - laplacian(u)

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## Customizations¶

### Customizing the symbols of the orthonormal frame vectors¶

By default, the vectors of the orthonormal frame associated with spherical coordinates are denoted $(e_r,e_\theta,e_\phi)$:

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frame = E.spherical_frame()
frame

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But this can be changed, thanks to the method set_name:

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frame.set_name('a', indices=('r', 'th', 'ph'),
latex_indices=('r', r'\theta', r'\phi'))
frame

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v.display()

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frame.set_name(('hr', 'hth', 'hph'),
latex_symbol=(r'\hat{r}', r'\hat{\theta}', r'\hat{\phi}'))
frame

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v.display()

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### Customizing the coordinate symbols¶

The coordinates symbols are defined within the angle brackets <...> at the construction of the Euclidean space. Above we did

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E.<r,th,ph> = EuclideanSpace(coordinates='spherical')


which resulted in the coordinate symbols $(r,\theta,\phi)$ and in the corresponding Python variables r, th and ph (SageMath symbolic expressions). Using other symbols, for instance $(R,\Theta,\Phi)$, is possible through the optional argument symbols of the function EuclideanSpace. It has to be a string, usually prefixed by r (for raw string, in order to allow for the backslash character of LaTeX expressions). This string contains the coordinate fields separated by a blank space; each field contains the coordinate’s text symbol and possibly the coordinate’s LaTeX symbol (when the latter is different from the text symbol), both symbols being separated by a colon (:):

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E.<R,Th,Ph> = EuclideanSpace(coordinates='spherical', symbols=r'R Th:\Theta Ph:\Phi')


We have then

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E.atlas()

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E.frames()

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E.spherical_frame()

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v = E.vector_field(R*sin(2*Ph)*sin(Th)^2 + R,
R*sin(2*Ph)*sin(Th)*cos(Th),
2*R*cos(Ph)^2*sin(Th), name='v')
v.display()

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