variable

Examples of variable in the following topics:

• Functions of Several Variables

  • Multivariable calculus is the extension of calculus in one variable to calculus in more than one variable.
  • Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus in more than one variable : the differentiated and integrated functions involve multiple variables, rather than just one.
  • Functions with independent variables corresponding to each of the degrees of freedom are often used to model these systems, and multivariable calculus provides tools for characterizing the system dynamics.
  • We have also studied theorems linking derivatives and integrals of single variable functions.
  • Extensions of concepts used for single variable functions may require caution.

• Differentials

  • Differentials are the principal part of the change in a function $y = f(x)$ with respect to changes in the independent variable.
  • The precise meaning of the variables $dy$ and $dx$ depends on the context of the application and the required level of mathematical rigor.
  • In physical applications, the variables $dx$ and $dy$ are often constrained to be very small ("infinitesimal").
  • Higher-order differentials of a function $y = f(x)$ of a single variable $x$ can be defined as follows:
  • When the independent variable $x$ itself is permitted to depend on other variables, then the expression becomes more complicated, as it must also include higher-order differentials in $x$ itself.

• Change of Variables

  • One makes a change of variables to rewrite the integral in a more "comfortable" region, which can be described in simpler formulae.
  • One makes a change of variables to rewrite the integral in a more "comfortable" region, which can be described in simpler formulae.
  • When changing integration variables, however, make sure that the integral domain also changes accordingly.
  • There exist three main "kinds" of changes of variable (one in $R^2$, two in $R^3$); however, more general substitutions can be made using the same principle.
  • Use a change a variables to rewrite an integral in a more familiar region

• Partial Derivatives

  • A partial derivative of a function of several variables is its derivative with respect to a single variable, with the others held constant.
  • A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).
  • Suppose that f is a function of more than one variable.
  • To find the slope of the line tangent to the function at $P(1, 1, 3)$ that is parallel to the $xz$-plane, the $y$ variable is treated as constant.
  • The partial derivative of $f$ at the point $a = (a_1, \cdots, a_n) \in U$ with respect to the $i$th variable  is defined as:

• Probability

  • Probability density function describes the relative likelihood, or probability, that a given variable will take on a value.
  • Integration is commonly used in statistical analysis, especially when a random variable takes a continuum value.
  • In probability theory, a probability density function (pdf), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.
  • The probability for the random variable to fall within a particular region is given by the integral of this variable's probability density over the region.
  • For a continuous random variable $X$, the probability of $X$ to be in a range $[a,b]$ is given as:

• Linear Equations

  • A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.
  • A common form of a linear equation in the two variables $x$ and $y$ is:
  • Since terms of linear equations cannot contain products of distinct or equal variables, nor any power (other than $1$) or other function of a variable, equations involving terms such as $xy$, $x^2$, $y^{\frac{1}{3}}$, and $\sin x$ are nonlinear.
  • The parametric form of a linear equation involves two simultaneous equations in terms of a variable parameter $t$, with the following values:

• Optimization in Several Variables

  • An optimization process that involves only a single variable is rather straightforward.
  • The same strategy applies for optimization with several variables.
  • In this atom, we will solve a simple example to see how optimization involving several variables can be achieved.

• Iterated Integrals

  • An iterated integral is the result of applying integrals to a function of more than one variable.
  • An iterated integral is the result of applying integrals to a function of more than one variable (for example $f(x,y)$ or $f(x,y,z)$) in such a way that each of the integrals considers some of the variables as given constants.
  • In this way, indefinite integration does not make much sense for functions of several variables.
  • While the antiderivatives of single variable functions differ at most by a constant, the antiderivatives of multivariable functions differ by unknown single-variable terms, which could have a drastic effect on the behavior of the function.
  • Use iterated integrals to integrate a function with more than one variable

• Limits and Continuity

  • A study of limits and continuity in multivariable calculus yields counter-intuitive results not demonstrated by single-variable functions.
  • A study of limits and continuity in multivariable calculus yields many counter-intuitive results not demonstrated by single-variable functions .
  • For example, there are scalar functions of two variables with points in their domain which give a particular limit when approached along any arbitrary line, yet give a different limit when approached along a parabola.
  • Continuity in single-variable function as shown is rather obvious.

• The Chain Rule

  • For a function $U$ with two variables $x$ and $y$, the chain rule is given as $\frac{d U}{dt} = \frac{\partial U}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial U}{\partial y} \cdot \frac{dy}{dt}$.
  • The chain rule above is for single variable functions $f(x)$ and $g(x)$.
  • However, the chain rule can be generalized to functions with multiple variables.
  • For example, consider a function $U$ with two variables $x$ and $y$: $U=U(x,y)$.
  • This relation can be easily generalized for functions with more than two variables.