FY12CE Mathematics key terms

The value a function approaches as its input approaches a chosen point, whether or not the function is defined there. It is the foundation for continuity, derivatives, and integrals.

A property of a function whose value at a point equals the limit of the function there. Informally, the graph has no break, jump, or hole at that point.

The instantaneous rate of change of a function with respect to its variable, defined as the limit of a difference quotient. Geometrically it gives the slope of the tangent line.

The property that a function has a well-defined derivative at a point. Differentiability at a point implies continuity there, but continuity does not imply differentiability.

An interior point of a function's domain where the derivative is zero or fails to exist. Such points are candidates for local maxima, minima, or inflection behavior.

The way a curve bends, determined by the sign of the second derivative. Positive second derivative means concave up; negative means concave down; a sign change marks an inflection point.

The signed area between a function's graph and the horizontal axis over an interval, defined as a limit of Riemann sums. It yields a number, unlike the indefinite integral.

The result linking differentiation and integration: differentiating an accumulated area function recovers the integrand, and a definite integral equals the change in an antiderivative.

An integral over an unbounded interval or of a function with an infinite discontinuity. It is defined by a limit and either converges to a finite value or diverges.

An infinite power series expansion of a smooth function about a point, built from the function's derivatives there. Truncating it gives polynomial approximations with controllable error.

The property of a sequence or series whose terms approach a finite limit or whose partial sums approach a finite sum. Divergence means no such finite limit exists.

The derivative of a multivariable function with respect to one variable while holding the others fixed. It measures sensitivity to that single input direction.

The vector of all first partial derivatives of a scalar function. It points in the direction of steepest increase and is perpendicular to level curves or surfaces.

The matrix of first partial derivatives of a vector-valued function, describing its best linear approximation. Its determinant measures local volume scaling under the transformation.

A set closed under addition and scalar multiplication, satisfying axioms such as associativity, commutativity, and existence of a zero vector and additive inverses.

A property of a set of vectors where no nontrivial linear combination equals the zero vector. Equivalently, no vector in the set is a combination of the others.

A linearly independent set of vectors that spans a vector space, so every element is expressed uniquely as a linear combination of them. The number of basis vectors is the dimension.

The dimension of the column space of a matrix, equal to the number of linearly independent rows or columns. It determines whether a linear system has unique, many, or no solutions.

A scalar computed from a square matrix that is nonzero exactly when the matrix is invertible. It also gives the signed volume scaling factor of the associated linear map.

A scalar for which some nonzero vector is mapped to a scalar multiple of itself by a linear operator. Eigenvalues characterize stability, vibration modes, and diagonalizability.

Writing a matrix as a product involving a diagonal matrix and a matrix of eigenvectors. It is possible exactly when there are enough linearly independent eigenvectors.

A scalar operation on two vectors equal to the sum of products of corresponding components. It relates to the cosine of the angle between them and tests orthogonality.

A vector operation defined in three dimensions, producing a vector perpendicular to both inputs with magnitude equal to the area of the spanned parallelogram.

The component of one vector along the direction of another, computed using the dot product. It underlies least squares fitting and Gram-Schmidt orthogonalization.

An equation relating an unknown function of one variable to its derivatives. Its order is the highest derivative present; solutions form families determined by initial conditions.

A differential equation together with specified values of the unknown function and possibly its derivatives at a single point. Under suitable conditions it has a unique solution.

An equation in which the unknown function and its derivatives appear only to the first power and are not multiplied together. Superposition of solutions then applies.

The algebraic equation obtained by substituting an exponential trial solution into a linear constant-coefficient differential equation. Its roots determine the solution's form and behavior.

The property that solutions of a system remain bounded or return to an equilibrium after small perturbations. For linear systems it is decided by the signs of eigenvalue real parts.

An integral transform converting a function of time into a function of a complex frequency variable. It turns linear differential equations into algebraic equations solved by partial fractions.

An operation combining two functions by integrating the product of one with a reversed, shifted copy of the other. It models the output of a linear system from its input and impulse response.

A function assigning a numerical value to each outcome of a random experiment. It may be discrete, taking countable values, or continuous, described by a density.

The probability-weighted average value of a random variable, also called its mean. For continuous variables it is computed by integrating value times density.

The expected squared deviation of a random variable from its mean, measuring spread. Its square root is the standard deviation, expressed in the original units.

A nonnegative function whose integral over an interval gives the probability that a continuous random variable falls in that interval. It integrates to one over the whole line.

A function giving the probability that a random variable is at most a given value. It is nondecreasing, ranges from zero to one, and its derivative is the density when continuous.

The result that sums or averages of many independent, identically distributed variables tend toward a normal distribution as sample size grows, regardless of the underlying distribution.

A range of plausible values for an unknown parameter, constructed so that a stated proportion of repeated samples would produce intervals containing the true value.

A procedure comparing observed data against a null hypothesis using a test statistic. A small p-value indicates the data are unlikely under the null.

A fitting method choosing parameters that minimize the sum of squared residuals. For linear models the solution follows from normal equations involving matrix transposes.

A declarative statement that is either true or false, never both. Compound propositions are built from simpler ones using connectives such as and, or, and not.

A compound statement of the form if P then Q, false only when P is true and Q is false. Its contrapositive is logically equivalent to it.

The statement formed by negating and swapping the parts of an implication, so if P then Q becomes if not Q then not P. It is logically equivalent to the original.

A symbol expressing how many elements satisfy a predicate, typically universal (for all) or existential (there exists). Negating one switches it to the other and negates the predicate.

A proof technique establishing a base case and then showing that truth for one case implies truth for the next. It concludes the statement holds for all natural numbers from the base.

An equation defining each term of a sequence from earlier terms. Solving one means finding a closed form, often by iteration, characteristic roots, or generating functions.

A formal power series whose coefficients encode the terms of a sequence. Algebraic manipulation of the series often yields closed forms and counts of combinatorial objects.

A collection of vertices together with edges joining pairs of them. Graphs model networks; properties such as connectivity, cycles, and coloring are central objects of study.

A connected graph containing no cycles. Any two vertices are joined by exactly one path, and a tree on n vertices has exactly n minus one edges.

A closed walk traversing every edge of a graph exactly once. Such a circuit exists exactly when the graph is connected and every vertex has even degree.

The property that an algorithm does not unduly amplify rounding errors introduced during computation. Unstable algorithms can produce wildly inaccurate results from tiny perturbations.

A measure of how strongly a problem's output responds to small relative changes in its input. Large values mean the problem is ill-conditioned and results are sensitive.

The error introduced by replacing an exact mathematical process, such as a limit or infinite series, with a finite approximation. It is distinct from rounding error.

An iterative root-finding scheme using the tangent line at the current estimate to produce the next one. It converges quadratically near a simple root but can fail otherwise.

Approximating a definite integral using weighted sums of function values at sample points, as in trapezoidal or Simpson rules. Accuracy depends on step size and smoothness.

Constructing a function that passes exactly through a given set of data points. Polynomial interpolation is common but can oscillate badly at interval edges.

A simplified mathematical description of a real system, built from assumptions, variables, and relationships. Its validity is judged by how well predictions match observations.

A state of a model where the governing equations give zero rate of change, so the system stays constant. Its stability determines whether nearby states return or diverge.

The study of how changes in a model's parameters or assumptions affect its outputs. It identifies which inputs matter most and how robust conclusions are.

The process of finding the inputs that minimize or maximize an objective function, possibly subject to constraints. Calculus, linear algebra, and numerical methods all supply tools.