algebraic

Examples of algebraic in the following topics:

• Further Transcendental Functions

  • A transcendental function is a function that is not algebraic.
  • A transcendental function is a function that "transcends" algebra in the sense that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, power, and root extraction.
  • In dimensional analysis, transcendental functions are notable because they make sense only when their argument is dimensionless (possibly after algebraic reduction).
  • One could attempt to apply a logarithmic identity to get $\log(10) + \log(m)$, which highlights the problem: applying a non-algebraic operation to a dimension creates meaningless results.
  • Identify a transcendental function as one that cannot be expressed as the finite sequence of an algebraic operation

• Finding Limits Algebraically

  • For a real-valued function expressed in terms of other functions, limit values may be computed via algebraic operations.
  • If $f$ is a real-valued (or complex-valued) function, then taking the limit is compatible with the algebraic operations, provided the limits on the right sides of the equations below exist (the last identity holds only if the denominator is non-zero).
  • This set of rules is often called the algebraic limit theorem, expressed formally as follows:

• Four Ways to Represent a Function

  • Note that in an algebraic representation, the input number is represented as a variable (in this case, an x).
  • One of the most important skills in algebra and calculus is being able to convert a function between these different forms, and this theme will recur in different forms throughout the text.

• Vectors in Three Dimensions

  • A Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction and can be added to other vectors according to vector algebra.
  • The coordinate representation of vectors allows the algebraic features of vectors to be expressed in a convenient numerical fashion.

• The Cross Product

  • The algebraic method of finding the cross product of two vectors involves inputting the vector information into matrices and manipulating them:

• Linear and Quadratic Functions

  • In calculus and algebra, the term linear function refers to a function that satisfies the following two linearity properties:
  • Linear functions form the basis of linear algebra.

• Indeterminate Forms and L'Hôpital's Rule

  • In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits.
  • Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is known as an indeterminate form.

• Conic Sections

  • In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2.

• Cylinders and Quadric Surfaces

  • In coordinates $\{x_1, x_2, \cdots, x_{D+1} \}$, the general quadric is defined by the algebraic equation:

• Newton's Method

  • The idea of the method is as follows: one starts with an initial guess which is reasonably close to the true root, then the function is approximated by its tangent line (which can be computed using the tools of calculus), and one computes the $x$-intercept of this tangent line (which is easily done with elementary algebra).