Ch. 4 Key Terms - Calculus Volume 3 | OpenStax

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Key Terms

boundary point: a point $P_{0}$ of $R$ is a boundary point if every $δ$ disk centered around $P_{0}$ contains points both inside and outside $R$

closed set: a set $S$ that contains all its boundary points

connected set: an open set $S$ that cannot be represented as the union of two or more disjoint, nonempty open subsets

constraint: an inequality or equation involving one or more variables that is used in an optimization problem; the constraint enforces a limit on the possible solutions for the problem

contour map: a plot of the various level curves of a given function $f (x, y)$

critical point of a function of two variables

the point

(x_{0}, y_{0})

is called a critical point of

f (x, y)

if one of the two following conditions holds:

$f_{x} (x_{0}, y_{0}) = f_{y} (x_{0}, y_{0}) = 0$
At least one of $f_{x} (x_{0}, y_{0})$ and $f_{y} (x_{0}, y_{0})$ do not exist

differentiable: a function $f (x, y, z)$ is differentiable at $(x_{0}, y_{0})$ if $f (x, y)$ can be expressed in the form $f (x, y) = f (x_{0}, y_{0}) + f_{x} (x_{0}, y_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}) (y - y_{0}) + E (x, y),$
where the error term $E (x, y)$ satisfies $lim_{(x, y) \to (x_{0}, y_{0})} \frac{E (x, y)}{\sqrt{{(x - x_{0})}^{2} + {(y - y_{0})}^{2}}} = 0$

directional derivative: the derivative of a function in the direction of a given unit vector

discriminant: the discriminant of the function $f (x, y)$ is given by the formula $D = f_{x x} (x_{0}, y_{0}) f_{y y} (x_{0}, y_{0}) - {(f_{x y} (x_{0}, y_{0}))}^{2}$

function of two variables: a function $z = f (x, y)$ that maps each ordered pair $(x, y)$ in a subset $D$ of $ℝ^{2}$ to a unique real number $z$

generalized chain rule: the chain rule extended to functions of more than one independent variable, in which each independent variable may depend on one or more other variables

gradient: the gradient of the function $f (x, y)$ is defined to be $\nabla f (x, y) = (\partial f / \partial x) i + (\partial f / \partial y) j,$ which can be generalized to a function of any number of independent variables

graph of a function of two variables: a set of ordered triples $(x, y, z)$ that satisfies the equation $z = f (x, y)$ plotted in three-dimensional Cartesian space

higher-order partial derivatives: second-order or higher partial derivatives, regardless of whether they are mixed partial derivatives

interior point: a point $P_{0}$ of $R$ is a boundary point if there is a $δ$ disk centered around $P_{0}$ contained completely in $R$

intermediate variable: given a composition of functions (e.g., $f (x (t), y (t))),$ the intermediate variables are the variables that are independent in the outer function but dependent on other variables as well; in the function $f (x (t), y (t)),$ the variables $x and y$ are examples of intermediate variables

Lagrange multiplier: the constant (or constants) used in the method of Lagrange multipliers; in the case of one constant, it is represented by the variable $λ$

level curve of a function of two variables: the set of points satisfying the equation $f (x, y) = c$ for some real number $c$ in the range of $f$

level surface of a function of three variables: the set of points satisfying the equation $f (x, y, z) = c$ for some real number $c$ in the range of $f$

linear approximation: given a function $f (x, y)$ and a tangent plane to the function at a point $(x_{0}, y_{0}),$ we can approximate $f (x, y)$ for points near $(x_{0}, y_{0})$ using the tangent plane formula

method of Lagrange multipliers: a method of solving an optimization problem subject to one or more constraints

mixed partial derivatives: second-order or higher partial derivatives, in which at least two of the differentiations are with respect to different variables

objective function: the function that is to be maximized or minimized in an optimization problem

open set: a set $S$ that contains none of its boundary points

optimization problem: calculation of a maximum or minimum value of a function of several variables, often using Lagrange multipliers

partial derivative: a derivative of a function of more than one independent variable in which all the variables but one are held constant

partial differential equation: an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives

region: an open, connected, nonempty subset of $ℝ^{2}$

saddle point: given the function $z = f (x, y),$ the point $(x_{0}, y_{0}, f (x_{0}, y_{0}))$ is a saddle point if both $f_{x} (x_{0}, y_{0}) = 0$ and $f_{y} (x_{0}, y_{0}) = 0,$ but $f$ does not have a local extremum at $(x_{0}, y_{0})$

surface: the graph of a function of two variables, $z = f (x, y)$

tangent plane: given a function $f (x, y)$ that is differentiable at a point $(x_{0}, y_{0}),$ the equation of the tangent plane to the surface $z = f (x, y)$ is given by $z = f (x_{0}, y_{0}) + f_{x} (x_{0}, y_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}) (y - y_{0})$

total differential: the total differential of the function $f (x, y)$ at $(x_{0}, y_{0})$ is given by the formula $d z = f_{x} (x_{0}, y_{0}) d x + f_{y} (x_{0}, y_{0}) d y$

tree diagram: illustrates and derives formulas for the generalized chain rule, in which each independent variable is accounted for

vertical trace: the set of ordered triples $(c, y, z)$ that solves the equation $f (c, y) = z$ for a given constant $x = c$ or the set of ordered triples $(x, d, z)$ that solves the equation $f (x, d) = z$ for a given constant $y = d$

$δ$ ball: all points in $ℝ^{3}$ lying at a distance of less than $δ$ from $(x_{0}, y_{0}, z_{0})$

$δ$ disk: an open disk of radius $δ$ centered at point $(a, b)$

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Authors: Gilbert Strang, Edwin “Jed” Herman
Publisher/website: OpenStax
Book title: Calculus Volume 3
Publication date: Mar 30, 2016
Location: Houston, Texas
Book URL: https://openstax.org/books/calculus-volume-3/pages/1-introduction
Section URL: https://openstax.org/books/calculus-volume-3/pages/4-key-terms

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