### Key Terms

- boundary point
- a point ${P}_{0}$ of $R$ is a boundary point if every $\delta $ 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 $f\left(x,y\right)$

- critical point of a function of two variables
- the point $\left({x}_{0},{y}_{0}\right)$ is called a critical point of $f\left(x,y\right)$ if one of the two following conditions holds:

- ${f}_{x}\left({x}_{0},{y}_{0}\right)={f}_{y}\left({x}_{0},{y}_{0}\right)=0$
- At least one of ${f}_{x}\left({x}_{0},{y}_{0}\right)$ and ${f}_{y}\left({x}_{0},{y}_{0}\right)$ do not exist

- differentiable
- a function $f\left(x,y,z\right)$ is differentiable at $\left({x}_{0},{y}_{0}\right)$ if $f\left(x,y\right)$ can be expressed in the form $f(x,y)=f({x}_{0},{y}_{0})+{f}_{x}({x}_{0},{y}_{0})\left(x-{x}_{0}\right)+{f}_{y}({x}_{0},{y}_{0})\left(y-{y}_{0}\right)+E\left(x,y\right),$

where the error term $E\left(x,y\right)$ satisfies $\underset{\left(x,y\right)\to \left({x}_{0},{y}_{0}\right)}{\text{lim}}\frac{E\left(x,y\right)}{\sqrt{{\left(x-{x}_{0}\right)}^{2}+{\left(y-{y}_{0}\right)}^{2}}}=0$

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

- discriminant
- the discriminant of the function $f\left(x,y\right)$ is given by the formula $D={f}_{xx}({x}_{0},{y}_{0}){f}_{yy}({x}_{0},{y}_{0})-{\left({f}_{xy}({x}_{0},{y}_{0})\right)}^{2}$

- function of two variables
- a function $z=f\left(x,y\right)$ that maps each ordered pair $\left(x,y\right)$ in a subset $D$ of ${\mathbb{R}}^{2}$ 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 $f\left(x,y\right)$ is defined to be $\nabla f\left(x,y\right)=\left(\partial f\text{/}\partial x\right)i+\left(\partial f\text{/}\partial y\right)j,$ 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 $\left(x,y,z\right)$ that satisfies the equation $z=f\left(x,y\right)$ 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 $\delta $ disk centered around ${P}_{0}$ contained completely in $R$

- intermediate variable
- given a composition of functions (e.g., $f\left(x\left(t\right),y\left(t\right)\right)),$ the intermediate variables are the variables that are independent in the outer function but dependent on other variables as well; in the function $f\left(x\left(t\right),y\left(t\right)\right),$ the variables $x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y$ 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 $\lambda $

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

- linear approximation
- given a function $f\left(x,y\right)$ and a tangent plane to the function at a point $\left({x}_{0},{y}_{0}\right),$ we can approximate $f\left(x,y\right)$ for points near $\left({x}_{0},{y}_{0}\right)$ 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 ${\mathbb{R}}^{2}$

- saddle point
- given the function $z=f(x,y),$ the point $\left({x}_{0},{y}_{0},f\left({x}_{0},{y}_{0}\right)\right)$ is a saddle point if both ${f}_{x}\left({x}_{0},{y}_{0}\right)=0$ and ${f}_{y}\left({x}_{0},{y}_{0}\right)=0,$ but $f$ does not have a local extremum at $\left({x}_{0},{y}_{0}\right)$

- surface
- the graph of a function of two variables, $z=f\left(x,y\right)$

- tangent plane
- given a function $f\left(x,y\right)$ that is differentiable at a point $\left({x}_{0},{y}_{0}\right),$ the equation of the tangent plane to the surface $z=f\left(x,y\right)$ is given by $z=f\left({x}_{0},{y}_{0}\right)+{f}_{x}\left({x}_{0},{y}_{0}\right)\left(x-{x}_{0}\right)+{f}_{y}\left({x}_{0},{y}_{0}\right)\left(y-{y}_{0}\right)$

- total differential
- the total differential of the function $f\left(x,y\right)$ at $\left({x}_{0},{y}_{0}\right)$ is given by the formula $dz={f}_{x}\left({x}_{0},{y}_{0}\right)dx+{f}_{y}\left({x}_{0},{y}_{0}\right)dy$

- 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 $\left(c,y,z\right)$ that solves the equation $f\left(c,y\right)=z$ for a given constant $x=c$ or the set of ordered triples $\left(x,d,z\right)$ that solves the equation $f\left(x,d\right)=z$ for a given constant $y=d$

- $\delta $ ball
- all points in ${\mathbb{R}}^{3}$ lying at a distance of less than $\delta $ from $\left({x}_{0},{y}_{0},{z}_{0}\right)$

- $\delta $ disk
- an open disk of radius $\delta $ centered at point $\left(a,b\right)$