Glossary of Terms

circulation

simply the line integral of a vector field around a closed path

concavity

measures whether a function of one variable opens upward (concavity positive, second derivative positive,) downward (concavity and second derivative negative,) or is flat (all zero)

constraint

usually a function that describes a required domain for a function to be optimized; think of budget constraints as applied to producing goods

conservative vector field

a conservative vector field is a vector with the property that the line integral of this vector field between any two points is path-independent

critical point

a point on the graph of a function is called a critical point if either the derivatives of the function are zero or non-existent at this point

curl

the curl of a vector field is a vector which points in the direction normal to the plane of greatest circulation density and has a magnitude equal to the circulation density in this direction

curl field

these vector fields can be written as the curl of another vector field, which is called the vector potential

curvature

closely related to the second derivative(s) of a function

del operator

this is the derivative operator which (1) takes a scalar function into its vector gradient, (2) operates through the divergence to take a vector field into its scalar divergence, and (3) operates as the curl to produce the vector curl of a vector field

derivative

derivatives are taken with respect to a particular variable and measure the rate of change of the value of the function as the variable increases

domain

the is the set of objects which a function acts on

directional derivative

this is the rate of change of a function in the direction of a vector

divergence

this quantity measures the tendency (at a point) of a vector field to spread out (positive divergence) or squish in (negative divergence)

extrema

an extrema is a point on the graph of a function where the value of the function is either (1) larger than all points around it, or (2) smaller than all points around it

flux

by calculating the flux of a vector field through a surface, one finds the volume rate at which a fluid (whose velocity is given by the vector field) is flowing through the surface

function

a relationship between two sets called the domain and range; functions uniquely associate each element of the domain with a single element of the range

global minimum

a global minimum is a point, P, on a function, f, so that for every point Q (in the domain) which is not P, f(Q) > f(P); a function can have several global minima, but the value of the function is the same at each one

global maximum

a global minimum is a point, P, on a function, f, so that for every point Q (in the domain) which is not P, f(Q) < f(P); a function can have several global maxima, but the value of the function is the same at each one

gradient

the gradient of a function of several variables is a vector with the properties (1) its direction at a point is always toward the maximum rate of increase of the function, and (2) its magnitude at a point is equal to the maximum rate of increase of the function at the point

gradient field

any vector field which can be written as the gradient of a scalar function (called the scalar potential function)

inflection point

this is a point along the graph of a function of one variable where the concavity changes

Lagrange multiplier

in a constrained optimization problem,this is the constant of proprtionality between the gradients of the constraint and the optimized function; it represents "how much bang for your buck" or the rate of change of the optimum value of the function as the constraint changes

local linearity

this is the idea that, on very small scales (ie. locally) any smooth function looks like a plane and thus has local properties of a plane; you are most familiar withthis on the earth's surface: even though the surface is curved, locallay (on our scale of a few miles of sight) it appears flat

local minimum

this is a point, P, on a function, f, such that for all points Q in a small neighborhood of P, f(Q) > f(P); a function can have many local minima with different values of the function at each

local maximum

this is a point, P, on a function, f, such that for all points Q in a small neighborhood of P, f(Q) < f(P); a function can have many local maxima with different values of the function at each

nabla

this is another name for the del operator

optimization

the process of finding (1) where a function has maximum or minimum values, and (2) what those maximum and minimum values are

oriented curve

any curve (one dimensional object) in which a direction of positive velocity has been chosen unambiguously

oriented surface

a surface on which a direction of positive flow (and thus flux) has been chosen; this amounts to specifying the normal vector to the surface

path independent field

a vector field is path indepent if every path, C, between the fixed points P and Q produces the same value for the line integral of the vector field along C; path-independent vector fields can be shown to be gradient fields

potential function

a potential function is the scalar valued function whose gradient gives rise to a gradient or path independent vector field

range

this is the set which a function takes elements of the range into

right hand rule

this refers to defining the direction of a positive cross product; pointing your right index finger along a vector A and your right middle finger toward B and then extending your thumb will ensure that your thumb points in the direction of the vector A x B

saddle point

this is a critical point, P, on a function, f, so that there are points Q1 and Q2 nearby with f(Q1) > f(P) and f(Q2) < f(P)

scalar

this is an object which holds only one piece of information (how big it is)

second derivative test

there are two versions of this: (1) for single variable functions, check the concavity at a critical point to determine whether the critical point is a max or min, (2) for multivariable functions f(x,y) there is a test involoving D = f_xx f_yy - f_xy² which determines the classification of the critical point

smooth

a function is said to be smooth if any order derivative exists for any value in the domain of the function

solenoidal

a vector field which can be written as the curl of another vector field (the vector potential)

vector

an object which stores multiple pieces of information; often considered as having a magnitude (length of an arrow) and a direction

vector field

a function whose domain and range are both sets of vector quantities

vector potential

this is the vector field which gives rise to a solenoidal field through the curl operator