How do symbolic computations work

Symbolic computation involves manipulating mathematical expressions using symbols (like variables) rather than specific numbers, allowing for general solutions and analysis of patterns within equations, essentially solving problems by treating variables as abstract representations without assigning fixed values, unlike numerical computations which deal with concrete numbers; this is achieved through algorithms that can perform operations like simplification, substitution, differentiation, and integration on these symbolic expressions, providing deeper insights into the mathematical relationships involved. [1, 2, 3, 4, 5]

Key aspects of symbolic computation: [2, 3, 4]

Symbolic variables: Instead of using fixed numbers, symbolic computations use variables that represent unknown values, enabling manipulation of formulas without needing specific numerical input. [2, 3, 4]
Expression manipulation: Algorithms within a computer algebra system (CAS) can perform operations like expanding, factoring, simplifying, and combining expressions based on mathematical rules. [1, 2, 6]
Symbolic differentiation and integration: Complex calculus operations like differentiation and integration can be performed on symbolic expressions, allowing for analytical solutions to differential equations. [1, 2, 7]
Substitution: Replacing variables with other expressions within a formula is a fundamental operation in symbolic computation, enabling analysis of different scenarios. [2, 8, 9]

Example: [2, 3, 4]

Imagine wanting to find the derivative of the function f(x) = x^2 + 2x. [2, 3, 4]

Numerical approach: Plug in a specific value for x, calculate the function's value, then slightly change x and calculate again to approximate the derivative.
Symbolic approach: A symbolic computation software would treat "x" as a symbol and use differentiation rules to directly calculate the derivative as 2x + 2, providing a general expression applicable to any value of x.

Applications of symbolic computation: [1, 7, 9]

Physics and engineering: Analyzing complex systems by manipulating equations representing physical phenomena. [1, 7, 9]
Mathematics research: Exploring mathematical properties and proving theorems through symbolic manipulation. [1, 2, 3]
Control systems design: Deriving transfer functions and analyzing system behavior using symbolic expressions. [1, 7, 9]
Computer graphics: Generating geometric shapes and transformations using symbolic calculations. [1, 5, 9]