Howard Anton

On the application of distribution theory to variational problems

Abstract The fixed end-point problem in variational calculus for an extremal containing corners is considered in a distributional context. The distributional Euler equation obtained is valid over the entire extremal and implicitly contains the Weierstrass-Erdmann corner conditions.

COUNTING TECHNIQUES; PERMUTATIONS AND COMBINATIONS

This chapter presents counting techniques, permutations, and combinations. There are two kinds of counting problems—the problem of counting the number of elements in a set, and the problem of counting the number of ways that the elements of a set can be arranged. The arrangements of objects are called permutations if the order is important. The arrangements of objects are called combinations if the order is not important. The general counting rule is called the multiplication principle. The multiplication principle can be extended to more than two decisions. An arrangement of a set of distinct objects in a definite order without repetitions is called a permutation of the set.

FUNCTIONS, LIMITS, AND RATES OF CHANGE

This chapter discusses the functions, limits, and rates of change. The study of calculus is concerned with the mathematical study of change. There are two basic geometric problems that call for the use of calculus: (a) finding a tangent to a curve, and (b) finding the area under a curve. There is a close relationship between the tangent problem and the problem of determining the rate at which a variable quantity is changing in value. The portion of calculus concerned with this problem is commonly called differential calculus. The portion of calculus concerned with these ideas is commonly called integral calculus. A function that is continuous at every point is called continuous; if there is at least one point of discontinuity, then the function is called discontinuous.

FUNCTIONS OF SEVERAL VARIABLES

This chapter discusses the functions of several variables. A rectangular coordinate system in three dimensions, also called a Cartesian coordinate system, is formed by three mutually perpendicular coordinate lines that intersect at their origins. The point of intersection of the axes is called the origin of the coordinate system. Each pair of coordinate axes determines a plane, called a coordinate plane. These are referred to as the xy plane , the xz plane , and the yz plane . To each point P in three-dimensional space, one can assign an ordered triple of numbers ( a, b, c ), called the coordinates of P . It can be done by passing three planes through P parallel to the coordinate planes and recording the coordinates a , b , and c of the intersections of the planes with the x , y , and z axis, respectively. For functions of two variables, there are analogous notions.

APPLICATIONS OF DIFFERENTIATION

This chapter discusses the applications of differentiation and explains that differentiation can be used to solve certain kinds of maximization and minimization problems. One of the important problems in any retail business is inventory control. On the one hand, there must be enough inventory to meet demand, and on the other hand, the business must avoid excess inventory as this results in unnecessary storage, insurance, and management costs. Moreover, money not committed to inventory can be the earning interest elsewhere. As an example, suppose a retailer of automobile tires expects to sell 8000 tires during the year with sales occurring at a fairly constant rate. Although the retailer could order the 8000 tires all at once, this would result in high holding costs, such as insurance, storage rental, security, and so forth. To reduce the holding costs the retailer might, instead, make many smaller orders during the year. However, this result in high reorder costs, such as delivery charges, paperwork, loading and unloading, and so forth. This leads to the problem of determining an ordering strategy that will strike an optimal balance between holding costs and reorder costs. More precisely, the problem is to minimize by choosing an appropriate lot size. The lot size that minimizes the total annual inventory cost is called the economic ordering quantity.

MATHEMATICS OF FINANCE

This chapter discusses mathematical methods and formulas that are useful in business and personal finance. For many transactions, interest is added to the principal at regular time intervals so that the interest itself earns interest. This is called compounding of interest. The time interval between successive additions of interest is called the conversion period. The more frequent the compounding, the greater the total interest. The terms principal, amount, future value, and present value have the same meaning in compound interest problems that they have in simple interest problems. For a fixed principal, time period, and annual interest rate, the more frequent the compounding, the greater will be the return on the investment. In continuous compounding, at each instant of time the investment grows in proportion to its current value. The money deducted in advance is called the discount and the money received by the borrower is called the proceeds. The time period between successive annuity payments is called the payment period or payment interval for the annuity.

FOUR – MATRICES AND LINEAR SYSTEMS

Publisher Summary This chapter presents a study of systems which may involve more than two equations or more than two unknowns. A linear system that has no solutions is said to be inconsistent; if it has at least one solution, it is called consistent. The rows, also called horizontal lines, of an augmented matrix correspond to the equations in the associated linear system. A rectangular array of numbers is called a matrix. The numbers in the array are called the entries of the matrix. There are four properties of an augmented matrix in reduced row echelon form. Those four properties are: (1) if a row is not made up entirely of zeros, then the leftmost nonzero number in the row is a 1; (2) if there are any rows consisting entirely of zeros, they are all together at the bottom of the matrix; (3) in two successive rows, not consisting entirely of zeros, the first nonzero number in the lower row is to the right of the first non-zero number in the upper row; and (4) each column that contains the first nonzero number of some row has zeros everywhere else.

THREE – LINEAR PROGRAMMING (A GEOMETRIC APPROACH)

Publisher Summary This chapter presents a relatively new area of mathematics called linear programming. The word programming is derived from the early applications of the subject to problems in the programming or allocation of supplies. The transportation problem is an important type of linear programming problem. In this kind of problem there are various warehouses and various stores and the cost of shipping an item from any warehouse to any store is known. A convex set means connecting any two points in the set by a line segment to put the line segment completely in the set. The convex sets are of two types: bounded and unbounded. A bounded set is one that can be enclosed by some suitably large circle, and an unbounded set is one that cannot be so enclosed. A corner point in a convex set is any point in the set that is the intersection of two boundary lines. An optimal solution is a feasible solution that makes the objective function as large or as small as possible.

APPLICATIONS OF INTEGRATION

Publisher Summary This chapter discusses the applications of the definite integral to problems in business, economics, motion, biology, consumption of natural resources, and probability. These applications are based on an important relationship between the change in the value of a function and the definite integral of its derivative. The marginal cost (MC), marginal revenue (MR), and marginal profit (MP) are the derivatives of the cost function, revenue function, and profit function, respectively. An equation that involves an unknown function and its derivatives is called a differential equation. The differential equations arise in a variety of important applications, especially those involving rates of change. A function that satisfies a differential equation is called a solution of that equation. A condition that specifies the value of y for some specific value of x is called an initial condition, and the problem of solving a differential equation subject to an initial condition is called an initial value problem.

FIVE – LINEAR PROGRAMMING (AN ALGEBRAIC APPROACH)

Publisher Summary The simplex method is applicable only when a linear programming problem satisfies certain conditions. This chapter discusses a simplified version of these conditions as well as certain other preliminaries. A standard linear programming problem means that the objective function is to be maximized, the variables are all required to be non-negative, and in the other constraints, the expressions involving the variables are less than or equal to ( ≤ ) a non-negative constant. The row and column containing the pivot entry are called, respectively, the pivot row and pivot column. The new implicit variable is usually called the departing variable as it departs from the set of explicit variables and becomes implicit. There are certain difficulties that occur when the minimum non-negative quotient, which is computed to determine the departing variable, occurs in two or more rows. This situation is called degeneracy.