multivariable

(adjective)

concerning more than one variable

Related Terms

Examples of multivariable 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.
    • Multivariable calculus can be applied to analyze deterministic systems that have multiple degrees of freedom.
    • Quantitative analysts in finance also often use multivariate calculus to predict future trends in the stock market.
    • As we will see, multivariable functions may yield counter-intuitive results when applied to limits and continuity.

  • 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 .
    • Continuity in each argument does not imply multivariate continuity.
    • However, continuity in multivariable functions yields many counter-intuitive results.
    • Describe the relationship between the multivariate continuity and the continuity in each argument

  • Applications of Minima and Maxima in Functions of Two Variables

    • Finding extrema can be a challenge with regard to multivariable functions, requiring careful calculation.
    • We have learned how to find the minimum and maximum in multivariable functions.
    • As previously mentioned, finding extrema can be a challenge with regard to multivariable functions.
    • Identify steps necessary to find the minimum and maximum in multivariable functions

  • Maximum and Minimum Values

    • The second partial derivative test is a method in multivariable calculus used to determine whether a critical point $(a,b, \cdots )$ of a function $f(x,y, \cdots )$ is a local minimum, maximum, or saddle point.

  • The Chain Rule

    • The chain rule can be used to take derivatives of multivariable functions with respect to a parameter.

  • Directional Derivatives and the Gradient Vector

    • The directional derivative of a multivariate differentiable function along a given vector $\mathbf{v}$ at a given point $\mathbf{x}$ intuitively represents the instantaneous rate of change of the function, moving through $\mathbf{x}$ with a velocity specified by $\mathbf{v}$.

  • Double Integrals Over Rectangles

    • Let us assume that we wish to integrate a multivariable function $f$ over a region $A$:

  • Iterated Integrals

    • 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.

  • Fundamental Theorem for Line Integrals

    • If $\varphi$ is a differentiable function from some open subset $U$ (of $R^n$) to $R$, and if $r$ is a differentiable function from some closed interval $[a,b]$ to $U$, then by the multivariate chain rule, the composite function $\circ r$ is differentiable on $(a,b)$ and $\frac{d}{dt}(\varphi \circ \mathbf{r})(t)=\nabla \varphi(\mathbf{r}(t)) \cdot \mathbf{r}'(t)$ for for all $t$ in $(a,b)$.

  • Tangent Planes and Linear Approximations

    • For a surface given by a differentiable multivariable function $z=f(x,y)$, the equation of the tangent plane at $(x_0,y_0,z_0)$ is given as: