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Calculus and Analytic Geometry [Seventh Edition]

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A calculus textbook containing exercises and problem solving strategies to help the student better grasp the different techniques offered.

Hardcover

First published January 1, 1920

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Displaying 1 - 26 of 26 reviews
Profile Image for Scribble Orca.
213 reviews397 followers
October 16, 2012
It got me through first year university. That makes it five stars.
Profile Image for Superconformal Hassaan.
65 reviews22 followers
June 2, 2021
Calculus and Analytical Geometry by Thomas and Finney was the first book that I read after I started my undergraduate studies. I was so captivated by this book that I read it from cover to cover. I had learned calculus before when I was in O levels but this book presented the foundations of calculus and the application of the various techniques very well.

The book starts with a chapter on preliminaries as any good undergraduate-level calculus or mathematical methods book should start (Arfken, Weber, and Harris also start with a chapter on preliminaries). This chapter majorly deals with functions, their properties, and their graphs. Moreover, this chapter deals with the rules for manipulating and reading graphs. This chapter also devotes some time to discuss trigonometry and the related identities.

Then, the second chapter is devoted to understanding the concept of limits and the notion of continuity. Here, you will find the epsilon-delta stuff that mathematicians love to talk about.

After these two introductory chapters, the next two chapters are devoted to differentiation and their applications. The third chapter explains the techniques required to do differentiation (which aren't a lot in number as differentiation is easy) and derives the derivatives of elementary functions from first principles. The techniques such as chain rule, product rule (and the Leibniz rule) are discussed. In the fourth chapter, the application of differentiation in the optimization problems, calculating limits (i.e. the L'Hopitals's rule), proving theorems (e.g. mean value theorem), and numerical analysis (i.e. Newton Raphson method) are discussed.

The following two chapters focus on Integration and its applications. The fifth chapter establishes the notion of the Reimann integral (which is the mathematical name of the integral that most of the physics and engineering students do but it's not the only type of integral. There is also a class of integrals called Lebesgue integrals). This chapter also establishes the definite integral as the area between two curves. The fundamental theorem of calculus is also proven in the fifth chapter. The sixth chapter focuses on the applications of definite integral in calculating a plethora of quantities in physics (like the center of mass, the moment of inertia, and work done by a force) and in mathematics (like the centroid of a shape, volumes, and areas of revolution about an axis).

The seventh chapter is about the non-algebraic elementary functions (also known as the transcendental function) that include the hyperbolic, logarithmic, exponential, inverse hyperbolic, trigonometric, and inverse trigonometric functions. This chapter deals with the integration techniques when these functions are in the integrand. The techniques involve the relevant substitutions and the plethora of very interesting problems make the learning of these techniques very engaging.

In the eighth chapter, the authors discuss the notion of infinite series and how to make sense of infinite series. They discuss the standard convergence tests, including integral, ratio, root, and comparison tests. They also establish the notion of alternating and absolute convergences. Moreover, power series are discussed and it is here, where the authors discuss the notion of the McLaurin and Taylor series. Equipped with the testing techniques developed in the chapter, the authors establish the criteria for a power series to converge, and thus, the notion of a radius of convergence is established. The applications of the power series are discussed after these topics.

In the ninth chapter, the authors introduce the notion of conic sections which is truly a very useful concept to have for some very important physics problems e.g. Kepler problem and Rutherford scattering problem. The concept of eccentricity and the classification of the conic sections based on eccentricity is also done in the first half of this chapter. The second half of this chapter introduces the notion of polar coordinates and the relevant techniques regarding the polar coordinates e.g. graphing in the polar coordinates. Lastly, the form of conic section equations are derived in polar coordinates (as they are easy to analyze when the conic sections arise in the context of the Kepler problem).

The tenth chapter introduces vectors and the relevant concepts like dot and cross products. In the second half of this chapter, the concept of analytical geometry is introduced and the equations of lines, planes, and quadric surfaces are derived (i.e. cylinders, hyperboloids, ellipsoids, etc).

The eleventh chapter uses the concept of vectors derived in the previous chapter to analyze mechanical problems. A brief theory of space curves (i.e. three-dimensional curves) is also discussed with the notion of tangent, normal, and torsion vectors. Moreover, a useful discussion of Kepler's problems and derivations of Kepler's laws are also found in this chapter.

The last three chapters are directed towards multivariable calculus. The twelfth chapter discusses the techniques involved with the analysis of functions with more than one variable. The topics that go into such a discussion are chain rule for multivariable functions (that follows from Euler's increment theorem), higher dimensional optimization problems that lead to the notions of saddle points, and the relevant technique of Lagrange multipliers.

The thirteenth chapter is about doing multivariable integrals in some scalar fields in various important coordinate systems. These integrals are crucial in physics because they are required to calculate the masses, moments, and many more quantities. The subtle topic of substitutions in multiple integrals is also discussed as this topic requires some extra attention because students may make a mistake in doing such a substitution.

The fourteenth and the last chapter of this book is about the integration techniques in the presence of vector fields and this topic is far richer than its scalar analog. The relevant powerful theorems like divergence theorem, stoke's theorem, green's theorem, and important concepts like potentials, and conservative force fields are discussed. Moreover, the techniques to parametrize a surface (to calculate surface integrals) are also discussed with relevant examples (i.e. sphere, cone, and cylinder). This chapter has a rather detailed discussion regarding Stoke's theorem and divergence theorem with proofs of these theorems for different cases and interesting examples. For example, Stoke's theorem is discussed for polyhedral surfaces and surfaces with holes in separate sections. This chapter is crucial for the future study of electromagnetism and thus, I would recommend that any serious student of physics should spend ample time on this chapter.

The textbook has more than enough problems for a student to practice. Many of the problems require similar skills to solve but there are more structured questions in many exercises that have relevance to a physics or mathematics problem or a concept from these fields of study. I would recommend spending more time on these problems and I won't recommend solving all of the problems in the exercise set that require similar skills as it might be a waste of time. However, everyone learns at a different pace, and thus, if someone thinks that he/she needs to solve more problems that are similar to each other to grasp a concept, then listen to your inner voice first and do as many problems as you need to understand a particular concept or skill.

I hope this was helpful. Happy reading.
Profile Image for Nia.
Author 3 books195 followers
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August 13, 2017
I cannot rate this book against my previous text books because I have not gone through the entire book, but I do like page 574, as I recall with some amount of nostalgia learning to solve partial differential equations in my Theromdynamics class before havaing taken Differential Equations, and thinking that it was so much better that way, and wishing that all application subjects could be taught before the pure subject classes, rather than as we do it the other way around.
Profile Image for Amanda.
333 reviews
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May 26, 2010
Just brushing up...don't ask me why...I don't know!!
1 review
February 3, 2019
I read the book when I was studying in Engineering. This is the best book of Analytical Calculus I have ever read in my life!!!!!!
Specially I was fascinated by the portion of the book on curve tracing!!!!!!
Profile Image for Brey Laude.
25 reviews3 followers
April 7, 2022
If you need Calculus for your engineering or science work, Thomas is a natural choice. There are over 12 editions, however, and there are better choices among these myriad options depending upon what type of problems occupy your research or field of interest.
Profile Image for Bumbles.
269 reviews26 followers
July 22, 2023
Our 200-student class had a final exam average of 20 percent. Prof (brilliant btw) had pulled some graduate-level calc exams from the MIT archives, and I'm convinced it is because of our friend George here.
1 review
ñoble
January 27, 2021
It app is good and hope it's going to help me
This entire review has been hidden because of spoilers.
Displaying 1 - 26 of 26 reviews

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