This Dover reprint of the 1968 edition offers a valuable learning aid for calculus students who have the fundamentals of differentiation and integration behind them and are on the verge of PDEs. Here the focus is on developing a geometric intuition. Visualization on the plane is the “two-dimensional” aspect. I particularly appreciate the author’s patient and enlightening multi-method explanation of key topics. For instance, the classic and illustrative case of minimum area for a fixed perimeter rectangle is carefully detailed in approaches with level curves, implicit differentiation, Lagrange multipliers, and more. Similarly, a key example illustrating Taylor expansion is handled with multiple approaches.
The book begins with vector basics as a natural launching point for the geometric focus. In the final chapters focusing on the double integral, this planar thinking will be especially enlightening for students nonplussed by centroids, moments, etc. Of course, the assessment of the area of a closed curve is the book’s introduction of the integral and this naturally leads to the double integral representing volume with appropriate illustrations continuing the geometric theme. This approach is comparable to that taken in Maak′s An Introduction to Modern Calculus (1963) which I feel is also worthy of a reprint edition. Each chapter concludes with exercises; for many, answers are provided in the back of the book. This is a self-contained work excellent for self-study or as an adjunct text for the undergraduate approaching this level of calculus.
Tom Schulte prepares students for calculus at Oakland Community College in Auburn Hills, MI.