Symbolic mathematics involves the manipulation of mathematical symbols rather than numeric calculations.Unlike numerical mathematics, which focuses on approximating solutions using numbers, symbolic mathematics provides exact solutions. This approach ensures that the results are expressed in a form that preserves all inherent mathematical relationships. By working with symbolic expressions, one can perform algebraic manipulations, simplifications, and exact evaluations. This is particularly useful in fields such as algebra, calculus, and differential equations, where maintaining the precise mathematical structure is crucial. Symbolic mathematics tools, like Maxima-CAS, Yacas, SymPy, and Mathematica, empower users to derive exact answers, manipulate equations, and gain deeper insights into the underlying mathematical concepts.
Example:
Maxima helps in performing symbolic mathematics by automating the manipulation of mathematical symbols. These systems can handle a wide range of tasks, from simple algebraic manipulations to complex calculus problems.
To differentiate (or integrate) a function symbolically, you would typically define the function as a symbolic expression and then apply a differentiation (or integration) function.
R is open-source (free) software designed for statistical computing and graphics, but it is not primarily intended for symbolic mathematics.
To use symbolic mathematics with R, you need to install R from CRAN. Then, install the Ryacas library1, which provides an interface to the Yacas computer algebra system (CAS).
library(Ryacas)
# Define the function f(x) = x^2 - x - 6 using ysym from the Ryacas package
f <- ysym("x^2 - x - 6")
# Differentiate the function f with respect to x
deriv(f, "x")y: 2*x-1# Integrate the function f with respect to x
integrate(f, "x")y: x^3/3-x^2/2-6*x
Python is an open-source (free) programming language widely used for various types of programming and software development, but it is not primarily designed for symbolic mathematics.
To perform symbolic mathematics with Python, you need to install Python from Python.org. Then, install the SymPy library2, which provides capabilities for symbolic computation.
# Importing necessary functions from the sympy library
from sympy import symbols, diff, integrate
# Defining a symbolic variable 'x'
x = symbols('x')
# Defining the function f(x) = x^2 + 3x + 5
f = x**2 + 3*x + 5
# Calculating the derivative of f with respect to x
diff_f = diff(f, x)
# Displaying the derivative
diff_f2*x + 3# Calculating the indefinite integral of f with respect to x
int_f = integrate(f, x)
# Displaying the integral
int_fx**3/3 + 3*x**2/2 + 5*x
MATLAB is a programming and numeric computing platform that requires a paid license (which can be quite expensive). However, at ORU, you will have access to MATLAB during your education. For more information, visit MathWorks MATLAB.
% Define a symbolic variable 'x'
syms x
% Define the function f(x) = x^2
f = x^2;
% Differentiate the function f with respect to x
f_prime = diff(f, x);
% Integrate the function f with respect to x
f_integral = int(f, x);
% Display the original function f(x)
disp(['f(x) = ', char(f)]);
% Display the derivative f'(x)
disp(['f''(x) = ', char(f_prime)]);
% Display the integral of f(x)
disp(['Integral of f(x) = ', char(f_integral)]);
Mathematica is a programming and numeric computing platform that requires a paid license (which can be quite expensive). For more information, visit Wolfram Mathematica.
(* Define the function f with x as a variable.
The underscore _ after x indicates that x is a pattern variable,
meaning f can take any expression as an argument.
:= This is a delayed assignment operator in Mathematica,
meaning the right-hand side is evaluated each time f is called.*)
f[x_] := x^2
(* Differentiate the function f with respect to x.
D is the differentiation function in Mathematica. *)
fPrime = D[f[x], x]
(* Integrate the function f with respect to x.
Integrate is the integration function in Mathematica. *)
fIntegral = Integrate[f[x], x]
(* Print the original function f(x) *)
Print["f(x) = ", f[x]]
(* Print the derivative of f, f'(x) *)
Print["f'(x) = ", fPrime]
(* Print the integral of f, the indefinite integral of f(x) *)
Print["Integral of f(x) = ", fIntegral]
Maxima is a system for the manipulation of symbolic and numerical expressions, including differentiation, integration, Taylor series, Laplace transforms, ordinary differential equations, systems of linear equations, polynomials, sets, lists, vectors, matrices and tensors. Maxima yields high precision numerical results by using exact fractions, arbitrary-precision integers and variable-precision floating-point numbers. Maxima can plot functions and data in two and three dimensions.
The Maxima source code can be compiled on many systems, including Windows, Linux, and MacOS X. The source code for all systems and precompiled binaries for Windows and Linux are available at the SourceForge file manager.
(%i1) /* Define the function f(x) = x^5 + 2*x + 3 */
f: x^5 + 2*x + 3;(%o1) x^5+2*x+3(%i2) /* Differentiate the function f with respect to x */
diff(f, x);(%o2) 5*x^4+2(%i3) /* Integrate the function f with respect to x */
integrate(f, x);(%o3) x^6/6+x^2+3*x