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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
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