SymPy is a Python library used for symbolic mathematics. It allows you to manipulate mathematical expressions symbolically, solving equations, integrating functions, performing algebraic operations, and much more. In this tutorial, you will learn the basics of using SymPy for symbolic mathematics in Python.<\/p>\n
SymPy comes pre-installed with Anaconda, so if you are using Anaconda distribution, you already have SymPy installed. If you are using a different Python distribution or need to install SymPy for any other reason, you can install it using pip:<\/p>\n
pip install sympy\n<\/code><\/pre>\nTo verify the installation, open a Python interpreter or a Jupyter Notebook and import the SymPy library:<\/p>\n
import sympy\n<\/code><\/pre>\nIf no errors occur, the installation was successful.<\/p>\n
Creating Symbols<\/h2>\n
In SymPy, symbols are objects that represent mathematical symbols, such as variables or constants. To create a symbol, you can use the symbols()<\/code> function:<\/p>\nimport sympy as sp\n\n# Create a symbol\nx = sp.symbols('x')\n<\/code><\/pre>\nIn this example, we created a symbol x<\/code>. You can create multiple symbols at once by providing multiple symbol names as a string separated by spaces:<\/p>\nx, y, z = sp.symbols('x y z')\n<\/code><\/pre>\nManipulating Expressions<\/h2>\n
Once you have created symbols, you can use them to define mathematical expressions. SymPy provides various functions and operators to manipulate these expressions. Here are some basic examples:<\/p>\n
Addition and Subtraction<\/h3>\n# Create symbols\nx, y = sp.symbols('x y')\n\n# Define expressions\nexpr1 = x + y\nexpr2 = x - y\n\n# Print expressions\nprint(expr1) # x + y\nprint(expr2) # x - y\n<\/code><\/pre>\nMultiplication and Division<\/h3>\n# Create symbols\nx, y = sp.symbols('x y')\n\n# Define expressions\nexpr1 = x * y\nexpr2 = x \/ y\n\n# Print expressions\nprint(expr1) # x*y\nprint(expr2) # x\/y\n<\/code><\/pre>\nPowers<\/h3>\n# Create symbols\nx, y = sp.symbols('x y')\n\n# Define expressions\nexpr1 = x**2\nexpr2 = x**y\n\n# Print expressions\nprint(expr1) # x**2\nprint(expr2) # x**y\n<\/code><\/pre>\nSquare Root and Other Functions<\/h3>\n# Create symbols\nx = sp.symbols('x')\n\n# Define expressions\nexpr1 = sp.sqrt(x)\nexpr2 = sp.sin(x)\n\n# Print expressions\nprint(expr1) # sqrt(x)\nprint(expr2) # sin(x)\n<\/code><\/pre>\nSimplifying Expressions<\/h2>\n
SymPy provides a simplify()<\/code> function to simplify expressions. It applies various mathematical rules and simplifies the expression as much as possible.<\/p>\n# Create a symbol\nx = sp.symbols('x')\n\n# Define an expression\nexpr = (x**2 + 2*x + 1) \/ (x + 1)\n\n# Simplify the expression\nsimplified_expr = sp.simplify(expr)\n\n# Print the simplified expression\nprint(simplified_expr) # x + 1\n<\/code><\/pre>\nIn this example, the expression (x**2 + 2*x + 1) \/ (x + 1)<\/code> is simplified to x + 1<\/code>.<\/p>\nSolving Equations<\/h2>\n
SymPy provides a powerful solve()<\/code> function to solve equations symbolically. It can solve linear and polynomial equations, transcendental equations, differential equations, and more.<\/p>\nLinear Equations<\/h3>\n# Create symbols\nx, y = sp.symbols('x y')\n\n# Define an equation\neqn = sp.Eq(2*x + 3*y, 10)\n\n# Solve the equation\nsol = sp.solve(eqn, (x, y))\n\n# Print the solutions\nprint(sol) # [{x: -3*y\/2 + 5}]\n<\/code><\/pre>\nIn this example, the equation 2*x + 3*y = 10<\/code> is solved for variables x<\/code> and y<\/code>. The solution is stored in a dictionary where the keys are the symbols and values are the solutions.<\/p>\nTranscendental Equations<\/h3>\n# Create a symbol\nx = sp.symbols('x')\n\n# Define an equation\neqn = sp.Eq(sp.sin(x) + sp.cos(x), 1)\n\n# Solve the equation\nsol = sp.solve(eqn, x)\n\n# Print the solutions\nprint(sol) # [pi\/4]\n<\/code><\/pre>\nIn this example, the equation sin(x) + cos(x) = 1<\/code> is solved for x<\/code>. The solution is a list of values, in this case, [pi\/4]<\/code>.<\/p>\nDifferential Equations<\/h3>\n# Create a symbol\nx = sp.symbols('x')\n\n# Define a differential equation\neqn = sp.Eq(sp.diff(x**2, x) - x, 0)\n\n# Solve the differential equation\nsol = sp.dsolve(eqn)\n\n# Print the solution\nprint(sol) # Eq(x**2\/2 - x**2*log(x) + C1, 0)\n<\/code><\/pre>\nIn this example, the differential equation d(x^2)\/dx - x = 0<\/code> is solved. The solution is x**2\/2 - x**2*log(x) + C1 = 0<\/code>, where C1<\/code> is an arbitrary constant.<\/p>\nDifferentiation and Integration<\/h2>\n
SymPy provides functions to differentiate and integrate expressions symbolically.<\/p>\n
Differentiation<\/h3>\n# Create a symbol\nx = sp.symbols('x')\n\n# Define an expression\nexpr = sp.sin(x)\n\n# Differentiate the expression\nderiv = sp.diff(expr, x)\n\n# Print the derivative\nprint(deriv) # cos(x)\n<\/code><\/pre>\nIn this example, the expression sin(x)<\/code> is differentiated with respect to x<\/code>. The derivative is cos(x)<\/code>.<\/p>\nIntegration<\/h3>\n# Create a symbol\nx = sp.symbols('x')\n\n# Define an expression\nexpr = sp.exp(x)\n\n# Integrate the expression\nintegral = sp.integrate(expr, x)\n\n# Print the integral\nprint(integral) # exp(x)\n<\/code><\/pre>\nIn this example, the expression exp(x)<\/code> is integrated with respect to x<\/code>. The integral is exp(x)<\/code>.<\/p>\nLimit Evaluation<\/h2>\n
SymPy allows you to evaluate limits of functions symbolically. The limit()<\/code> function is used to find the limit as a variable approaches a specific value.<\/p>\n# Create a symbol\nx = sp.symbols('x')\n\n# Define a function\nfunc = x \/ sp.sin(x)\n\n# Evaluate the limit\nlimit_value = sp.limit(func, x, 0)\n\n# Print the limit value\nprint(limit_value) # 1\n<\/code><\/pre>\nIn this example, the limit of x \/ sin(x)<\/code> as x<\/code> approaches 0<\/code> is evaluated. The limit value is 1<\/code>.<\/p>\nLinear Algebra<\/h2>\n
SymPy provides functions to perform basic linear algebra operations, such as matrix operations, solving linear systems, eigenvalues, eigenvectors, and more.<\/p>\n
Matrix Operations<\/h3>\n# Create symbols\nx, y, z = sp.symbols('x y z')\n\n# Create matrices\nA = sp.Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 10]])\nB = sp.Matrix([x, y, z])\n\n# Matrix multiplication\nC = A * B\n\n# Print the result\nprint(C) # Matrix([[x + 2*y + 3*z], [4*x + 5*y + 6*z], [7*x + 8*y + 10*z]])\n<\/code><\/pre>\nIn this example, matrix A<\/code> is multiplied by vector B<\/code>. The result is a new vector C<\/code>.<\/p>\nSolving Linear Systems<\/h3>\n# Create symbols\nx, y, z = sp.symbols('x y z')\n\n# Create a linear system\neqns = [sp.Eq(x + y + z, 1), sp.Eq(x - y + 2*z, 3), sp.Eq(2*x + y - z, 0)]\n\n# Solve the linear system\nsol = sp.solve(eqns, (x, y, z))\n\n# Print the solutions\nprint(sol) # {x: 1\/4, y: 1\/4, z: 1\/2}\n<\/code><\/pre>\nIn this example, a linear system of equations is solved for variables x<\/code>, y<\/code>, and z<\/code>. The solution is a dictionary where the keys are the symbols and values are the solutions.<\/p>\nEigenvalues and Eigenvectors<\/h3>\n# Create a symbol\nx = sp.symbols('x')\n\n# Create a matrix\nA = sp.Matrix([[1, 2], [3, x]])\n\n# Compute eigenvalues and eigenvectors\neigenvals = A.eigenvals()\neigenvects = A.eigenvects()\n\n# Print the results\nprint(eigenvals) # {x: sqrt(7)\/2 - 1\/2, sqrt(7)\/2 + 1\/2}\nprint(eigenvects) # [(sqrt(7)\/2 - 1\/2, 1, [[-(1\/3 - sqrt(7)\/6)], [1]]), (sqrt(7)\/2 + 1\/2, 1, [[-(1\/3 + sqrt(7)\/6)], [1]])]\n<\/code><\/pre>\n