critical, or stationary, point

A point at which a function attains a critical value, either a local extremum or an n-dimensional saddlepoint.  The term stationary point may be visualized by placing a perfectly balanced ball on a smooth curve or surface:  at a stationary point the ball is in static equilibrium and will not roll.

C²

Twice continuously differentiable.  A scalar-valued function of a vector variable is said to be of class C¹ when the first partial derivatives of the function exist and are continuous.  If these derivatives, in turn, have continuous partials, the function is said to be of class C².  A related notion is that of geometric continuity:  a function with continuous tangent is of class G¹, and a function with continuous curvature is of class G².

degenerate critical point

A critical point at which the Hessian determinant is zero.  The behavior of a function near such points might be tested graphically with level sets (contours) and sections or by some other method.

discriminant

A relatively simple expression that determines some of the properties, as the nature of the roots, of a given equation or function.

gradient

The first derivative of a scalar-valued function of a vector variable.

Hessian determinant

The determinant of the Hessian matrix.

Hessian matrix

The second derivative of a scalar-valued function of a vector variable.  The Hessian matrix is the Jacobian matrix of the gradient.

indefinite matrix

Let x be an n-element column vector and x^T its transpose.  Then an n x n symmetric matrix M is indefinite if the quadratic form x^T M x < 0 for some x and > 0 for other x.

Jacobian matrix

The first derivative of a vector-valued function of a vector variable.

local maximum

A point at which a function attains a value that is a maximum relative to other points in the neighborhood.  If the value at this point is the largest value of the function, the point is a global, or absolute, maximum, and the function is said to be concave.

Appendix.  Glossary (continued)

local minimum

A point at which a function attains a value that is a minimum relative to other points in the neighborhood.  If the value at this point is the smallest value of the function, the point is a global, or absolute, minimum, and the function is said to be convex.

local, or relative, extremum

A point at which a function attains a value that is a maximum or minimum relative to other points in the neighborhood.  If the value at this point is the largest or smallest value of the function, the point is a global, or absolute, extremum.  Also called turning point.

negative definite matrix

Let x be an n-element column vector and x^T its transpose.  Then an n x n symmetric matrix M is negative definite if for all nonzero x,  the quadratic form x^T M x < 0.

negative semidefinite matrix

Let x be an n-element column vector and x^T its transpose.  Then an n x n symmetric matrix M is negative semidefinite if for all nonzero x, the quadratic form x^T M x < 0.

positive definite matrix

Let x be an n-element column vector and x^T its transpose.  Then an n x n symmetric matrix M is positive definite if for all nonzero x,  the quadratic form x^T M x > 0.

positive semidefinite matrix

Let x be an n-element column vector and x^T its transpose.  Then an n x n symmetric matrix M is positive semidefinite if for all nonzero x, the quadratic form x^T M x > 0.

saddlepoint

A critical point that is not a local extremum.  At an n-dimensional saddlepoint, the tangent hyperplane is horizontal, but the value of the function is neither a local minimum nor a local maximum.

strict, or isolated, local extremum

A local extremum at which the function value is strictly less than (for a strict local minimum) or strictly greater than (for a strict local maximum) the function value of nearby points. In 3-space, local extrema that are not isolated exhibit as level lines or contours.