3 SymPy Documentation Tutorials

SymPy Documentation Tutorials

Link: https://docs.sympy.org/latest/index.html

GitHub - https://github.com/sympy

Importing sympy: Use the following code -

import sympy as sp
sp.init_printing(use_unicode=True)

Tutorials

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Introduction file

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Gotchas file

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SymPy Features

Documentation Link: https://docs.sympy.org/latest/tutorials/intro-tutorial/features.html

Basic Operations file

Documentation Link: https://docs.sympy.org/latest/tutorials/intro-tutorial/basic_operations.html

• Substitution (expr.subs)

• Converting strings to sympy expressions - sympify (sp.sympify)

• evalf (.evalf)

• lambdify (sp.lambdify)

Printing file

Features - Printing

Documentation Link: https://docs.sympy.org/latest/tutorials/intro-tutorial/printing.html

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Simplification file

Documentation Link: https://docs.sympy.org/latest/tutorials/intro-tutorial/simplification.html

• simplify: (sp.simplify)

• Polynomial/Rational Function Simplification

  • expand: (sp.expand)
  • factor: (sp.factor, sp.factor_list)
  • collect: (sp.collect, expr.coeff)
  • cancel: (sp.cancel)
  • apart: (sp.apart)

• Trigonometric Simplification

  • trigsimp (sp.trigsimp)
  • expand_trig (sp.expand_trig)

• Powers

  • powsimp (sp.powsimp)
  • expand_power_exp / expand_power_base (sp.expand_power_exp and sp.expand_power_base)
  • powdenest (sp.powdenest)

• Exponentials and logarithms

  • expand_log (sp.expand_log)
  • logcombine (sp.logcombine)

• Special Functions (sp.factorial, sp.bionomial, sp.gamma, sp.hyper)

  • rewrite (expr.rewrite)
  • expand_func (sp.expand_func)
  • hyperexpand (sp.hyperexpand)
  • combsimp (sp.combsimp)
  • gammasimp (sp.gammasimp)

• Example: Continued Fractions

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Calculus file

Documentation Link: https://docs.sympy.org/latest/tutorials/intro-tutorial/calculus.html

• Derivatives: (sp.diff() or expr.diff())
• Integrals: (sp.integrate)
• Limits: (sp.limit)
• Series Expansion: (expr.series())
• Finite differences: (sp.differentiate_finite(), expr.as_finite_difference(), sp.finite_diff_weights(), sp.apply_finite_diff ())

Solvers file

Documentation Link: https://docs.sympy.org/latest/tutorials/intro-tutorial/solvers.html

• A Note about Equations (sp.Eq, sp.solveset)

• Solving Equations Algebraically

  • System of linear equations (sp.linsolve)
  • Non-linear system of equations (sp.nonlinsolve)
  • Limitations (sp.solve)

• Solving Differential Equations (sp.dsolve)

Matrices file

Documentation Link: https://docs.sympy.org/latest/tutorials/intro-tutorial/matrices.html

Matrix in sympy: sp.Matrix

• Basic Operations
  • Shape (sp.shape, matrix.shape)
  • Accessing Rows and Columns (matrix.row, matrix.col)
  • Deleting and Inserting Rows and Columns (.row_del, .col_del, .row_insert, .col_insert)
• Basic Methods (.T)
• Matrix Constructors (sp.zeros, sp.ones, sp.diag)
• Advanced Methods
  • Determinant (.det)
  • RREF (.rref)
  • Nullspace (.nullspace)
  • Columnspace (.columnspace)
  • Eigenvalues, Eigenvectors, and Diagonalization (eigenvals, eigenvects, diagonalize)
• Possible Issues
  • Zero Testing

Advanced Expression Manipulation file

How-to Guides

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Explanations

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API References - Basics

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API References - Code Generation

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API References - Logic

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API References - Matrices

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Matrices (Linear Algebra) file

Documentation Link: https://docs.sympy.org/latest/modules/matrices/matrices.html

• Creating Matrices: (sp.Matrix(). Generating elements using functions.)
• Basic Manipulation
  • indexing and slicing
  • arithmetic operations
  • deleting rows and columns
  • vector operations (v1.cross(v2))
  • joining matrices (row_join, col_join)
• Operations on entries (matrix.applyfunc(), .subs)
• Linear algebra (matrix.inv())

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API References - Number Theory

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API References - Physics

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Documentation Link: https://docs.sympy.org/latest/reference/public/physics/index.html

Hydrogen Wavefunctions file

Documentation Link: https://docs.sympy.org/latest/modules/physics/hydrogen.html

Module: sympy.physics.hydrogen. Under this, the functions availabe are -

• E_nl(): (energy in Hartree atomic units)

• E_nl_dirac(): (relativistic energy)

• Psi_nlm(): (Total wavefunction $\psi_{nlm}(r,\phi,\theta)$)

• R_nl(): (Radial part of wavefunction $R_{nl}(r)$)

Matrices, Pauli Algebra file

Documentation Link: https://docs.sympy.org/latest/modules/physics/matrices.html

Module: sympy.physics.matrices. In this module the following functions are available -

• mgamma (Dirac gamma matrix $\gamma^\mu$)

• msigma(i) (Pauli matrix $\sigma_i$ for $i=1,2,3$)

• pat_matrix (Parallel Axis Theorem matrix to translate the inertia matrix a distance of $(dx,dy,dz)$ for a body of mass $m$.)

Documentation Link: https://docs.sympy.org/latest/modules/physics/paulialgebra.html

Module: sympy.physics.paulialgebra. In this module the following functions are available -

• Pauli
• evaluate_pauli_product

Quantum Harmonic Oscillator 1D and 3D file

Documentation Link: https://docs.sympy.org/latest/modules/physics/qho_1d.html

Module: sympy.physics.qho_1d. The following functions are available here -

• E_n (energy)
• coherent_state
• psi_n (wavefunction $\psi_n$)

Documentation Link: https://docs.sympy.org/latest/modules/physics/sho.html

Module: sympy.physics.sho. Following functions are available here -

• E_nl: (Energy $E_{nl} = (2n+l+\frac{3}{2})\hbar\omega$ of a 3d isotropic harmonic oscillator)

• R_nl: (Radial wavefunction $R_{nl}(r)$ of a 3d isotropic harmonic oscillator)

The Physics Vector Module file

Documentation Link: https://docs.sympy.org/latest/modules/physics/vector/index.html

To import this module, do use the following code:

import sympy as sp
sp.init_printing(use_unicode=True)

import sympy.physics.vector as spvec
spvec.init_vprinting(pretty_print=True)

• Vector and ReferenceFrame web link (.ReferenceFrame())

  • Vector Algebra (.dot, .cross, .normalize, .magnitude, .to_matrix)

  • Vector Calculus (.diff, .orient, .dcm, .orientnew, .dt)

• Kinematics web link (.dynamicsymbols(), .set_ang_vel, .ang_vel_in)

  • Multiple bodies with angular velocities defined relative to each other

  • Point (.Point, .locatenew, .pos_from, .set_vel, .vel, .set_acc, .acc)

  • Particle moving on a ring (.v2pt_theory, a2pt_theory)

• Potential Issues/Advanced Topics/Future Features in Physics/Vector Module web link

  • Inertia (import sympy.physics.mechanics, .inertia, .to_matrix)

  • ReferenceFrame (with indices and latexs)

  • dynamicsymbols

• Scalar and Vector Field Functionality web link

  • Fields web link (.express)

  • Field operators and other related functions web link

    • Curl (.curl)

    • divergence (.divergence)

    • gradient (.gradient)

    • Conservative and Solenoidal fields (.is_conservative, .is_solenoidal)

    • Scalar potential functions (.scalar_potential, .scalar_potential_difference)

• Physics Vector API web link

API References - Utilities

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API References - Topics

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