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 Df (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 = (xx, ...,  x). The set {P} of points Pi  = (x1i x2i , ..., xni ) such that f (P) = k, is called a level set. When f is a function of two variables, x and y, the graph of the points (xy) that satisfy f (xy) = 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 (xy) = x 2 + y 2. When f is a function of three variables, x, y, and z, the graph of the points (xyz) such that f (xyz) = 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 P(x,y), there are many distinct pairs of polar coordinates (r,) that locate the point. For example, if P(x,y) = P(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.