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 = fxx fyy - fxy2 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
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Copyright © 1998 by Kris H. Green
The Vector Calculus Website at
http://www.math.arizona.edu/~vector