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:

1. $f_x\left(x_0,y_0\right)=f_y\left(x_0,y_0\right)=0$
2. 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 ${ℝ}^2$ 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 $x\text{and}y$ 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 $\lambda$

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 ${ℝ}^2$

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 ${ℝ}^3$ 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 $\left(a,b\right)$