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Kernel: SageMath 10.1

Vector Calculus with SageMath

In [3]:

print(version())
%display latex

SageMath version 10.1, Release Date: 2023-08-20

The 3-dimensional Euclidean space

We can declare the 3D Euclidean space $\mathbb{E}^3$ , with $(x,y,z)$ as cartesian coordinates.

In [4]:

E.<x,y,z> = EuclideanSpace()
print(E)
E

Euclidean space E^3

$\displaystyle \mathbb{E}^{3}$

In [5]:

# Change the notations to i, j, and k, which is consistent with the textbook. 

frame = E.cartesian_frame()
frame.set_name(('hx', 'hy', 'hz'), 
               latex_symbol=(r'\mathbf{i}', r'\mathbf{j}', 
                             r'\mathbf{k}'))
frame

$\displaystyle \left(\mathbb{E}^{3}, \left(\mathbf{i},\mathbf{j},\mathbf{k}\right)\right)$

Vector fields

We define a vector field on $\mathbb{E}^3$ from its components in the vector frame $(\mathbf{i}, \mathbf{j}, \mathbf{k})$ .

In [6]:

# We define a vector field with three functions

v = E.vector_field(-y, x, sin(x*y*z), name='v')
v.display()

$\displaystyle v = -y \mathbf{i} + x \mathbf{j} + \sin\left(x y z\right) \mathbf{k}$

We can access to the components of the vector field $v$ via the square bracket operator.

In [7]:

print(v[1])
print(v[:])

-y
[-y, x, sin(x*y*z)]

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

(x,y,z)=(3,-2,1),\qquad v=2\mathbf{i}+3\mathbf{j}+\sin(-6)\mathbf{k}

In [9]:

p = E((3,-2,1), name='p')
print(p, ', whose coordiante are')
print(p.coordinates())

vp = v.at(p)
print(vp, 'is')
vp.display()

Point p on the Euclidean space E^3 , whose coordiante are
(3, -2, 1)
Vector v at Point p on the Euclidean space E^3 is

$\displaystyle v = 2 \mathbf{i} + 3 \mathbf{j} -\sin\left(6\right) \mathbf{k}$

Vector fields can be plotted (see the list of options for customizing the plot).

In [10]:

v.plot(max_range=1.5, scale=0.5, viewer='threejs', online=True)

Graphics3d Object

We may define a vector field with generic components.

In [11]:

u = E.vector_field(function('u_x')(x,y,z),
                   function('u_y')(x,y,z),
                   function('u_z')(x,y,z),
                   name='u')
print(u.display())
print(u[:])

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

u = u_x(x, y, z) hx + u_y(x, y, z) hy + u_z(x, y, z) hz
[u_x(x, y, z), u_y(x, y, z), u_z(x, y, z)]

$\displaystyle u = u_{x}\left(3, -2, 1\right) \mathbf{i} + u_{y}\left(3, -2, 1\right) \mathbf{j} + u_{z}\left(3, -2, 1\right) \mathbf{k}$

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:

In [12]:

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

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

$\displaystyle {u}\cdot{v}$

In [12]:

s.display()

$\displaystyle \begin{array}{llcl} {u}\cdot{v}:& \mathbb{E}^{3} & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & -y u_{x}\left(x, y, z\right) + x u_{y}\left(x, y, z\right) + \sin\left(x y z\right) u_{z}\left(x, y, z\right) \end{array}$

In [13]:

s(p)

$\displaystyle -\sin\left(6\right) u_{z}\left(3, -2, 1\right) + 2 \, u_{x}\left(3, -2, 1\right) + 3 \, u_{y}\left(3, -2, 1\right)$

In [14]:

s.expr()

$\displaystyle -y u_{x}\left(x, y, z\right) + x u_{y}\left(x, y, z\right) + \sin\left(x y z\right) u_{z}\left(x, y, z\right)$

Norm

The norm of a vector field is

In [15]:

s = norm(u)
print(s)
s.display()

Scalar field |u| on the Euclidean space E^3

$\displaystyle \begin{array}{llcl} \left\|u\right\|:& \mathbb{E}^{3} & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & \sqrt{u_{x}\left(x, y, z\right)^{2} + u_{y}\left(x, y, z\right)^{2} + u_{z}\left(x, y, z\right)^{2}} \end{array}$

In [16]:

s.expr()

$\displaystyle \sqrt{u_{x}\left(x, y, z\right)^{2} + u_{y}\left(x, y, z\right)^{2} + u_{z}\left(x, y, z\right)^{2}}$

In [17]:

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

$\displaystyle \mathrm{True}$

In [18]:

norm(v).expr()

$\displaystyle \sqrt{x^{2} + y^{2} + \sin\left(x y z\right)^{2}}$

Cross product

In [19]:

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

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

$\displaystyle {u}\times{v} = \left( \sin\left(x y z\right) u_{y}\left(x, y, z\right) - x u_{z}\left(x, y, z\right) \right) \mathbf{i} + \left( -\sin\left(x y z\right) u_{x}\left(x, y, z\right) - y u_{z}\left(x, y, z\right) \right) \mathbf{j} + \left( x u_{x}\left(x, y, z\right) + y u_{y}\left(x, y, z\right) \right) \mathbf{k}$

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.

In [15]:

w = E.vector_field(name='w')
w[1] = x*z
w[2] = y*z
w.display()

$\displaystyle w = x z \mathbf{i} + y z \mathbf{j}$

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

In [16]:

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

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

$\displaystyle -{\left(y u_{x}\left(x, y, z\right) - x u_{y}\left(x, y, z\right)\right)} z \sin\left(x y z\right) - {\left(x^{2} u_{z}\left(x, y, z\right) + y^{2} u_{z}\left(x, y, z\right)\right)} z$

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

In [21]:

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

True
True
True

Differential operators

While the standard operators grad, div, curl, etc. involved in vector calculus are accessible via the dot notation (e.g. v.div()), let us import functions grad, div, curl, etc. that allow for using standard mathematical notations (e.g. div(v)).

In [22]:

from sage.manifolds.operators import *

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 $(x,y,z)$ .

In [23]:

F = E.scalar_field(function('f')(x,y,z), name='F')
F.display()

$\displaystyle \begin{array}{llcl} F:& \mathbb{E}^{3} & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & f\left(x, y, z\right) \end{array}$

In [24]:

# The value of F at a point p

F(p)

$\displaystyle f\left(3, -2, 1\right)$

In [25]:

# The gradient of F

print(grad(F))
grad(F).display()

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

$\displaystyle \mathrm{grad}\left(F\right) = \frac{\partial\,f}{\partial x} \mathbf{i} + \frac{\partial\,f}{\partial y} \mathbf{j} + \frac{\partial\,f}{\partial z} \mathbf{k}$

In [26]:

# The norm of the gradient of F

norm(grad(F)).display()

$\displaystyle \begin{array}{llcl} \left\|\mathrm{grad}\left(F\right)\right\|:& \mathbb{E}^{3} & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & \sqrt{\left(\frac{\partial\,f}{\partial x}\right)^{2} + \left(\frac{\partial\,f}{\partial y}\right)^{2} + \left(\frac{\partial\,f}{\partial z}\right)^{2}} \end{array}$

Divergence

In [27]:

# The divergence of a vector field

s = div(u)
s.display()

$\displaystyle \begin{array}{llcl} \mathrm{div}\left(u\right):& \mathbb{E}^{3} & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & \frac{\partial\,u_{x}}{\partial x} + \frac{\partial\,u_{y}}{\partial y} + \frac{\partial\,u_{z}}{\partial z} \end{array}$

In [28]:

# The divergence of the vector field v = (-y,x,sin(xyz))

print(v.display())
div(v).expr()

v = -y hx + x hy + sin(x*y*z) hz

$\displaystyle x y \cos\left(x y z\right)$

In [29]:

# The divergence of the vector field w = (xz,yz)

print(w.display())
div(w).expr()

w = x*z hx + y*z hy

$\displaystyle 2 \, z$

In [30]:

# An identity valid for any scalar field F and any vector field u.

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

F: E^3 → ℝ
   (x, y, z) ↦ f(x, y, z)
u = u_x(x, y, z) hx + u_y(x, y, z) hy + u_z(x, y, z) hz

$\displaystyle \mathrm{True}$

Curl

In [31]:

# The curl of a vector field:

s = curl(u)
print(s)
s.display()

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

$\displaystyle \mathrm{curl}\left(u\right) = \left( -\frac{\partial\,u_{y}}{\partial z} + \frac{\partial\,u_{z}}{\partial y} \right) \mathbf{i} + \left( \frac{\partial\,u_{x}}{\partial z} - \frac{\partial\,u_{z}}{\partial x} \right) \mathbf{j} + \left( -\frac{\partial\,u_{x}}{\partial y} + \frac{\partial\,u_{y}}{\partial x} \right) \mathbf{k}$

In [32]:

# To use the notation rot instead of curl

from sage.manifolds.operators import curl as rot

In [33]:

# The curl of the vector field u

print(u.display())
rot(u).display()

u = u_x(x, y, z) hx + u_y(x, y, z) hy + u_z(x, y, z) hz

In [34]:

curl(v).display()

$\displaystyle \mathrm{curl}\left(v\right) = x z \cos\left(x y z\right) \mathbf{i} -y z \cos\left(x y z\right) \mathbf{j} + 2 \mathbf{k}$

In [35]:

curl(w).display()

$\displaystyle \mathrm{curl}\left(w\right) = -y \mathbf{i} + x \mathbf{j}$

In [36]:

# The curl of a gradient is always zero:

curl(grad(F)).display()

$\displaystyle \mathrm{curl}\left(\mathrm{grad}\left(F\right)\right) = 0$

In [37]:

# The divergence of a curl is always zero:

div(curl(u)).display()

$\displaystyle \begin{array}{llcl} \mathrm{div}\left(\mathrm{curl}\left(u\right)\right):& \mathbb{E}^{3} & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & 0 \end{array}$

In [38]:

# An identity valid for any scalar field F and any vector field u.

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

F: E^3 → ℝ
   (x, y, z) ↦ f(x, y, z)
u = u_x(x, y, z) hx + u_y(x, y, z) hy + u_z(x, y, z) hz

$\displaystyle \mathrm{True}$

Laplacian

In [39]:

# The Laplacian of a scalar field:

s = laplacian(F)
s.display()

$\displaystyle \begin{array}{llcl} \Delta\left(F\right):& \mathbb{E}^{3} & \longrightarrow & \mathbb{R} \\ & \left(x, y, z\right) & \longmapsto & \frac{\partial^2\,f}{\partial x ^ 2} + \frac{\partial^2\,f}{\partial y ^ 2} + \frac{\partial^2\,f}{\partial z ^ 2} \end{array}$

In [40]:

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

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

$\displaystyle \mathrm{True}$

In [41]:

# The Laplacian of a vector field:

laplacian(u).display()

$\displaystyle \Delta\left(u\right) = \left( \frac{\partial^2\,u_{x}}{\partial x ^ 2} + \frac{\partial^2\,u_{x}}{\partial y ^ 2} + \frac{\partial^2\,u_{x}}{\partial z ^ 2} \right) \mathbf{i} + \left( \frac{\partial^2\,u_{y}}{\partial x ^ 2} + \frac{\partial^2\,u_{y}}{\partial y ^ 2} + \frac{\partial^2\,u_{y}}{\partial z ^ 2} \right) \mathbf{j} + \left( \frac{\partial^2\,u_{z}}{\partial x ^ 2} + \frac{\partial^2\,u_{z}}{\partial y ^ 2} + \frac{\partial^2\,u_{z}}{\partial z ^ 2} \right) \mathbf{k}$

In [42]:

# In the Cartesian frame, the components of the Laplacian of a vector field are nothing but the Laplacians of the components of the vector field, as we can check:

e = E.cartesian_frame()
laplacian(u) == sum(laplacian(u[[i]])*e[i] for i in E.irange())

$\displaystyle \mathrm{True}$

In the above formula, u[[i]] return the $i$ -th component of $u$ as a scalar field, while $u[i]$ would have returned the coordinate expression of this scalar field; besides, $e$ is the Cartesian frame.

In [43]:

u[1]

$\displaystyle u_{x}\left(x, y, z\right)$

In [44]:

# For v and w, we have

laplacian(v).display()

$\displaystyle \Delta\left(v\right) = -{\left(x^{2} y^{2} + {\left(x^{2} + y^{2}\right)} z^{2}\right)} \sin\left(x y z\right) \mathbf{k}$

In [45]:

curl(curl(u)).display()

$\displaystyle \mathrm{curl}\left(\mathrm{curl}\left(u\right)\right) = \left( -\frac{\partial^2\,u_{x}}{\partial y ^ 2} - \frac{\partial^2\,u_{x}}{\partial z ^ 2} + \frac{\partial^2\,u_{y}}{\partial x\partial y} + \frac{\partial^2\,u_{z}}{\partial x\partial z} \right) \mathbf{i} + \left( \frac{\partial^2\,u_{x}}{\partial x\partial y} - \frac{\partial^2\,u_{y}}{\partial x ^ 2} - \frac{\partial^2\,u_{y}}{\partial z ^ 2} + \frac{\partial^2\,u_{z}}{\partial y\partial z} \right) \mathbf{j} + \left( \frac{\partial^2\,u_{x}}{\partial x\partial z} + \frac{\partial^2\,u_{y}}{\partial y\partial z} - \frac{\partial^2\,u_{z}}{\partial x ^ 2} - \frac{\partial^2\,u_{z}}{\partial y ^ 2} \right) \mathbf{k}$

In [46]:

grad(div(u)).display()

$\displaystyle \mathrm{grad}\left(\mathrm{div}\left(u\right)\right) = \left( \frac{\partial^2\,u_{x}}{\partial x ^ 2} + \frac{\partial^2\,u_{y}}{\partial x\partial y} + \frac{\partial^2\,u_{z}}{\partial x\partial z} \right) \mathbf{i} + \left( \frac{\partial^2\,u_{x}}{\partial x\partial y} + \frac{\partial^2\,u_{y}}{\partial y ^ 2} + \frac{\partial^2\,u_{z}}{\partial y\partial z} \right) \mathbf{j} + \left( \frac{\partial^2\,u_{x}}{\partial x\partial z} + \frac{\partial^2\,u_{y}}{\partial y\partial z} + \frac{\partial^2\,u_{z}}{\partial z ^ 2} \right) \mathbf{k}$

In [47]:

# another identity

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

$\displaystyle \mathrm{True}$

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Vector Calculus with SageMath

The 3-dimensional Euclidean space

Vector fields

Algebraic operations on vector fields

Dot product

Norm

Cross product

Scalar triple product

Differential operators

Gradient of a scalar field

Divergence

Curl

Laplacian

Product

Resources

Company

Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more, all in one place. Commercial Alternative to JupyterHub.

Vector Calculus with SageMath

The 3-dimensional Euclidean space

Vector fields

Algebraic operations on vector fields

Dot product

Norm

Cross product

Scalar triple product

Differential operators

Gradient of a scalar field

Divergence

Curl

Laplacian

Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more,
all in one place. Commercial Alternative to JupyterHub.