Contour Map: Let be a set of points in the domain of f and let
K = a set of the corresponding values in the range of f. Then the set of level curves,
, for , constitutes a contour map of f.
Directional Derivative: Suppose that the function f is differentiable at a point
P
n,
that OP is the position vector to P, and that u is a unit vector in
n. The directional derivative of f at P in the direction of
u, written Du f (P), is
The directional derivative is a scalar that gives the rate of change of f in the
direction of u. It may be calculated from the gradient of f
at P, , by rising the scalar or dot product
of the gradient vector
Level Curves:
One way on visualizing a function of two variables, , is to draw its level curves:
curves where , where k is restricted to real numbers. For example, for the
function represents a family of level
curves and
represents a specific level curve. The surface
is an elliptic paraboloid, , is a family of ellipses in the xy-plane,
and is a specific ellipse in this family.
If k denotes the height or elevation of the surface, , then the level
curves show where the graph of has height k. The level curves may be regarded as
the projections to the xy-plane of the traces of the graph of f in the horizontal plane z = k.
If you visualize the level curves as being lifted up to the surface at the indicated height, you can put together a picture
of the surface. The surface is steep where the level curves are close together and flatter where they are further apart.
Level Set: For any function f of n variables, define
W = f (P) where the point
P = (x1 , x2 , ...,
xn ).
The set {Pi } of points
Pi = (x1i ,
x2i , ..., xni )
such that f (Pi ) = k, is called a
level set.
When f is a function of two variables, x and y, the
graph of the points (x, y) that satisfy
f (x, y) = k
is called a level curve. For example, the graph of
x 2 + y 2 = 9
is a level curve in
2 for the function
f (x, y) = x 2 + y 2.
When f is a function of three variables, x, y, and z,
the graph of the points (x, y, z) such that
f (x, y, z) = k
is called a level surface in
3.
For example, the graph of the ellipsoid,
is a level surface for
Level Surface:
If W is a function, , whose domain is a set
of points in the range of f, if k is a number in the range of f, is an k is a number in the
range, of f, then the graph of the
equation is a three dimensional surface called
the level surface of the function at k.
If every point lies on the level surface ,
then the vector gradient is perpendicular (or normal) to every differential curve lying
in the surface and passing through .

Polar Coordinate System:
An alternative to the traditional rectangular
coordinate system to locate points in the plane. Let P be a point in the plane
with rectangular coordinates (x,y). The point P can be located in polar
coordinates (r,) by specifying both the distance
of OP, the radial line segment from the origin to P, and the positive
angle (measured counter-clockwise) between the positive x-axis and the radial line
segment r from the origin to P.
Relationships between the two coordinate systems can be obtained geometrically:
For a specified point
P0 (x0 ,y0 ),
there are many distinct pairs of polar coordinates
(r,) that locate the point. For example, if
P0 (x0 ,y0 ) = P0 (3,-3)
then P0 may be specified in polar coordinates by
or more generally, by
Spherical Coordinates:
Is a system of curvilinear coordinates in which the position of a point in space is
designated by its distance from the origin or pole, the angle
between the radius vector and a vertically directed polar axis,
called the cone angle or colatitude, and the angle
between the plane of and a fixed
meridian plane through the polar axis, called the polar angle or longitude.