In the first year of calculus we study limits, derivatives, and integrals of functions with a single input, and a single output. The transition to advanced calculus is made when we generalize the notion of “function” to something which may have multiple inputs and multiple outputs. In this more general context limits, derivatives, and integrals take on new meanings and have new geometric interpretations. For example, in first-year calculus the derivative represents the slope of a tangent line at a specified point. When dealing with functions of multiple variables there may be many tangent lines at a point, so there will be many possible ways to differentiate. The emphasis of this book is on developing enough familiarity with the material to solve difficult problems. Rigorous proofs are kept to a minimum. I have included numerous detailed examples so that you may see how the concepts really work. All exercises have detailed solutions that you can find at the end of the book. I regard these exercises, along with their solutions, to be an integral part of the material. The present work is suitable for use as a stand-alone text, or as a companion to any standard book on the topic. This material is usually covered as part of a standard calculus sequence, coming just after the first full year. Names of college classes that cover this material vary greatly. Possibilities include advanced calculus, multivariable calculus, and vector calculus. At schools with semesters the class may be called Calculus III. At quarter schools it may be Calculus IV. The best way to use this book is to read the material in each section and then try the exercises. If there is any exercise you don’t get, make sure you study the solution carefully. At the end of each chapter you will find a quiz to test your understanding. These short quizzes are written to be similar to one that you may encounter in a classroom, and are intended to take 20–30 minutes. They are not meant to test every idea presented in the chapter. The best way to use them is to study the chapter until you feel confident that you can handle anything that may be asked, and then try the quiz. You should have a good idea of how you did on it after looking at the answers. At the end of the text there is a final exam similar to one which you would find at the conclusion of a college class. It should take about two hours to complete. Use it as you do the quizzes. Study all of the material in the book until you feel confident, and then try it. Advanced calculus is an exciting subject that opens up a world of mathematics. It is the gateway to linear algebra and differential equations, as well as more advanced mathematical subjects like analysis, differential geometry, and topology. It is essential for an understanding of physics, lying at the heart of electro-magnetics, fluid flow, and relativity. It is constantly finding new use in other fields of science and engineering. I hope that the exciting nature of this material is conveyed here. - McGraw Hill Education