GSoC Final Report
This report summarizes the work done in my GSoC 2020 project, Implementation of Vector Integration with SymPy. Blog posts with step by step development of the project are available at friyaz.github.io.
About Me
I am Faisal Riyaz and I have completed my third year of B.Tech in Computer Engineering student from Aligarh Muslim University.
Overview
The goal of the project is to add functions and class structure to support the integration of scalar and vector fields over curves, surfaces, and volumes. My mentors were Francesco Bonazzi (Primary) and Divyanshu Thakur.
Work Completed
Here is a list of PRs which were opened during the span of GSoC:

(Merged) #19472: Adds
ParametricRegion
class to represent parametrically defined regions. 
(Merged) #19539: Adds
ParametricIntegral
class to represent integral of scalar or vector field over a parametrically defined region. 
(Merged) #19580: Modified API of
ParametricRegion
. 
(Merged) #19650: Added support to integrate over objects of geometry module.

(Merged) #19681: Added
ImplicitRegion
class to represent implicitly defined regions. 
(Megred) #19807: Implemented the algorithm for rational parametrization of conics.

(Merged) #20000: Added API of newly added classes to sympy’s documentation.

(Open) #19883: Allow
vector_integrate
to handleImplicitRegion
objects. 
(Open) #20021: Add usage examples.
We had an important discussion regarding our initial approach and possible problems on issue #19320
Defining Regions with their parametric equations
The ParametricRegion
class is used to represent a parametric region in space.
>>> from sympy.vector import CoordSys3D, ParametricRegion, vector_integrate
>>> from sympy.abc import r, theta, phi, t, x, y, z
>>> from sympy import pi, sin, cos, Eq
>>> C = CoordSys3D('C')
>>> circle = ParametricRegion((4*cos(theta), 4*sin(theta)), (theta, 0, 2*pi))
>>> disc = ParametricRegion((r*cos(theta), r*sin(theta)), (r, 0, 4), (theta, 0, 2*pi))
>>> box = ParametricRegion((x, y, z), (x, 0, 1), (y, 1, 1), (z, 0, 2))
>>> cone = ParametricRegion((r*cos(theta), r*sin(theta), r), (theta, 0, 2*pi), (r, 0, 3))
>>> vector_integrate(1, circle)
8*pi
>>> vector_integrate(C.x*C.x  8*C.y*C.x, disc)
64*pi
>>> vector_integrate(C.i + C.j, box)
(Integral(1, (x, 0, 1), (y, 1, 1), (z, 0, 2)))*C.i + (Integral(1, (x, 0, 1), (y, 1, 1), (z, 0, 2)))*C.j
>>> _.doit()
4*C.i + 4*C.j
>>> vector_integrate(C.x*C.i  C.z*C.j, cone)
9*pi
Integration over objects of geometry
module
Many regions like circle are commonly encountered in problems of vector calculus. the geometry module of SymPy already had classes for some of these regions. The functionparametric_region_list
was added to determine the parametric representation of such classes. vector_integrate
can directly integrate objects of Point, Curve, Ellipse, Segment and Polygon class of geometry
module.
>>> from sympy.geometry import Point, Segment, Polygon, Segment
>>> s = Segment(Point(4, 1, 9), Point(1, 5, 7))
>>> triangle = Polygon((0, 0), (1, 0), (1, 1))
>>> circle2 = Circle(Point(2, 3), 6)
>>> vector_integrate(6, s)
42
>>> vector_integrate(C.x*C.y, circle2)
72*pi
>>> vector_integrate(C.y*C.z*C.i + C.x*C.x*C.k, triangle)
C.z/2
Implictly defined Regions
In many cases, it is difficult to determine the parametric representation of a region. Instead, the region is defined using its implicit equation. To represent implicitly defined regions, we implemented the ImplicitRegion
class.
But to integrate over a region, we need to determine its parametric representation. the rationl_parametrization
method was added to ImplicitRegion
.
>>> from sympy.vector import ImplicitRegion
>>> circle3 = ImplicitRegion((x, y), (x4)**2 + (y+3)**2  16)
>>> parabola = ImplicitRegion((x, y), (y  1)**2  4*(x + 6))
>>> ellipse = ImplicitRegion((x, y), (x**2/4 + y**2/16  1))
>>> rect_hyperbola = ImplicitRegion((x, y), x*y  1)
>>> sphere = ImplicitRegion((x, y, z), Eq(x**2 + y**2 + z**2, 2*x))
>>> circle3.rational_parametrization()
(8*t/(t**2 + 1) + 4, 8*t**2/(t**2 + 1)  7)
>>> parabola.rational_parametrization()
(6 + 4/t**2, 1 + 4/t)
>>> rect_hyperbola.rational_parametrization(t)
(1 + (t + 1)/t, t)
>>> ellipse.rational_parametrization()
(t/(2*(t**2/16 + 1/4)), t**2/(2*(t**2/16 + 1/4))  4)
>>> sphere.rational_parametrization()
(2/(s**2 + t**2 + 1), 2*s/(s**2 + t**2 + 1), 2*t/(s**2 + t**2 + 1))
After PR #19883 gets merged, vector_integrate
can directly work upon ImplicitRegion
objects.
Future Work

In vector calculus, many regions cannot be defined using a single implicit equation. Instead, inequality or a combination of the implicit equation and conditional equations is required. For Example, a disc can only be represented using an inequality
ImplicitRegion(x**2 + y**2 < 9)
and a semicircle can be represented asImplicitRegion(x**2 + y**2 4) & (y < 0))
. Support for such implicit regions needs to be added. 
The parametrization obtained using the
rational_parametrization
method in most cases is a large expression. SymPy’sIntegral
is unable to work over such expressions. It takes too long and often does not returns a result. We need to either simplify the parametric equations obtained or fix Integral to handle them. 
Adding the support to plot objects of
ParamRegion
andImplicitRegion
Conclusion
This summer has been a great learning experience. Writing a functionality, testing, debugging it has given me good experience in testdriven development. I think I realized many of the goals that I described in my proposal, though there were some major differences in the plan and actual implementation. Personally, I hoped I could have done better. I could not work much on the project during the last phase due to the start of the academic year.
I would like to thank Francesco for his valuable suggestions and being readily available for discussions. He was always very friendly and helpful. I would love to work more with him. S.Y Lee also provided useful suggestions.
Special thanks to Johann Mitteramskogler, Professor Sonia Perez Diaz, and Professor Sonia L.Rueda. They personally helped me in finding the algorithm for determining the rational parametrization of conics.
SymPy is an amazing project with a great community. I would mostly stick around after the GSoC period as well and continue contributing to SymPy, hopefully, exploring the other modules as well.