This book is aimed at mathematics students, typically in the second year of a university course. The first chapter, however, is suitable for first-year students. Differentiable functions are treated initially from the standpoint of approximating a curved surface locally by a fiat surface. This enables both geometric intuition, and some elementary matrix algebra, to be put to effective use. In Chapter 2, the required theorems - chain rule, inverse and implicit function theorems, etc- are stated, and proved (for n variables), concisely and rigorously. Chapter 3 deals with maxima and minima, including problems with equality and inequality constraints. The chapter includes criteria for discriminating between maxima, minima and saddlepoints for constrained problems; this material is relevant for applications, but most textbooks omit it. In Chapter 4, integration over areas, volumes, curves and surfaces is developed, and both the change-of-variable formula, and the Gauss-Green-Stokes set of theorems are obtained. The integrals are defined with approximative sums (ex pressed concisely by using step-functions); this preserves some geometrical (and physical) concept of what is happening. Consequent on this, the main ideas of the 'differential form' approach are presented, in a simple form which avoids much of the usual length and complexity. Many examples and exercises are included.

In 1930, John B. Craven was born in Poona, India, to missionary parents. In 1935, the family returned to England in time for World War II and the austere years that followed. Living in a green belt around London, John learned to appreciate the beauties of the changing seasons as he played in a woodland nearby. In 1954, John earned a bachelor's degree from La Sierra College. From 1956-1962, his graduate studies at the University of Southern California led to a master's degree in English. In 1962, he married Shirley Stoft and started 30 years of full-time teaching at Glendale Community College. He helped introduce programmed and computerized instruction in the teaching of remedial composition. When he taught English and American literature survey classes, he sought to help his students understand and enjoy the value in literature. Since his retirement in 1992, he has continued to teach part-time at the same college.

The Longman GNVQ Business series provides coverage of all the mandatory units required at Foundation, Intermediate and Advanced level GNVQ. This text covers Advanced level and contains activities throughout to help students understand the issues covered, to apply the knowledge they have acquired and to encourage their own investigations.