Category: College Mathematics

Free Geometry and Topology Ebooks

College Mathematics, High School Mathematics, Resources and Freebies, Web 2.0 Technology

Below are free Geometry and Topology ebooks. Most of these books are in PDF format. Some of the links point to a website containing the ebooks, while some directly links to the pdf files. All of these ebooks are downloadable and free.

  1. CK-12 Geometry by V. Cifarelli, A. Gloag, D. Greenberg, J. Sconyers, B.
  2. Geometry for Elementary School by Wikibooks
  3. The Elements by Euclid

College Geometry and Beyond

  1. A First Course in Topology: Continuity and Dimension by John McCleary
  2. A Tour of Triangle Geometry by Paul Yiu
  3. Algebraic and Geometric Topology by Andrew Ranicki, Norman Levitt, Frank Quinn
  4. Algebraic Curves: an Introduction to Algebraic Geometry by William Fulton
  5. Algebraic geometry and projective differential geometry by Joseph M. Landsberg
  6. Algebraic Geometry by J.S. Milne
  7. An Introduction to Differentiable Manifolds and Riemannian Geometry by William M. Boothby
  8. An Introduction to Riemannian Geometry by Sigmundur Gudmundsson
  9. Classical Geometry by Danny Calegari
  10. Computational Geometry
  11. Course of Differential Geometry by Ruslan Sharipov
  12. Elementary Topology O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov
  13. Fractal Geometry by Michael Frame, Benoit Mandelbrot, Nial Neger
  14. Fundamentals of Geometry by Oleg A. Belyaev
  15. Geometric Asymptotics
  16. Geometry and Billiards by Serge Tabachnikov
  17. Geometry and Group Theory
  18. Geometry and Quantum Field Theory by Pavel Etingof
  19. Geometry Formulas and Facts by Silvio Levy
  20. Geometry study guide
  21. Geometry of Surfaces Nigel Hitchin
  22. Geometry of the Sphere by John C. Polking
  23. Geometry, Topology, Geometric Modeling by Jean Gallier
  24. Geometry Unbound Kiran S. Kedlaya
  25. Intrinsic Geometry of Surfaces by A. D. Aleksandrov and V. A. Zalgaller
  26. Introduction to Symplectic and Hamiltonian Geometry by Ana Cannas da Silva
  27. Natural Operations in Differential Geometry Ivan Kolar, Peter W. Michor, Jan Slovak
  28. Non-Euclidean Geometry by Henry Manning
  29. Open Problems in Topology by Jan Van Mill, George M. Reed
  30. Projective and Polar Spaces by Peter J. Cameron
  31. Projective Differential Geometry Old and New by V. Ovsienko, S. Tabachnikov
  32. Projective Geometry by Nigel Hitchin
  33. Riemann surfaces, dynamics and geometry Course Notes
  34. The Eightfold Way: The Beauty of Klein’s Quartic Curve editedy by Silvio Levy
  35. The Foundations of Geometry by David Hilbert
  36. The Geometry and Topology of Three-Manifolds William P Thurston
  37. The Geometry of Iterated Loop Spaces by J. P. May
  38. Topics in Finite Geometry: Ovals, Ovoids and Generalized Quadrangles by S. E. Payne
  39. Topology and Geometry for Physics by H. Eschrig
  40. Topology Without Tears by Sydney A. Morris

You may also want to view my list of Free Algebra and Free Calculus ebooks.

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Derivative and the Maximum Area Problem

Calculus and Analysis, College Mathematics, High School Calculus

Note: This is the third part of the Derivative Concept Series. The first part is The Algebraic and Geometric Meaning of Derivative and the second part is Derivative in Real Life Context.

Introduction

The computation of derivative is often seen in maximum and minimum problems.  In this article, we will discuss why do we get the derivative of a function and equate it to 0 when we want to get its maximum or minimum. To give you a concrete example, let us consider the problem below.

Find the maximum area a rectangle with perimeter 10 units.

Without using calculus, we can substitute values for the rectangle’s length, compute for its width and its corresponding area. If we set the interval to 0.5, then we can come up with the table shown in Figure 1.

Figure 1 - Table showing the length, width, and area of a rectangle with perimeter 10.

Looking at the table above, we can observe that a rectangle of length of 2.5, a square, has the maximum area. If we have prior calculus  knowledge, however, we know that whatever the value of our perimeter, a square having the given perimeter will always have the maximum area.

Using elementary algebra, if we let x be the width of our rectangle, it follows that the length is 5-x. Let f(x) be the area of the rectangle. In effect, the area of the rectangle is described by the equation f(x) = 5x - x^2. We want to maximize the area, which implies that we want to find the maximum value of f(x).

Figure 2 – A rectangle with Perimeter 10 and width x units.

In elementary calculus, to compute for the maximum value of f(x), we get its derivative, which is equal to 5 - 2x, which we will denote f'(x). We then equate the f'(x) to 0 resulting to the equation 5-2x=0 \Rightarrow x = 2.5 which is exactly the maximum value in the table above.

Derivative and Equation to 0

In the article the Algebraic and Geometric Meaning of Derivative, we have learned that the derivative of a function is the slope of the line tangent to that function at a particular point. From elementary algebra, we also have learned the properties of slopes. If a line is rising to the right, the slope is greater than 0; if the line is rising to the to the left, then the slope is less than 0. We have also learned that a horizontal line has slope 0 and the vertical line has an undefined slope.

Figure 3 – Properties of slope of a straight line.

In the problem above, we calculated by getting the derivative (the slope of the line tangent to a function at a particular point) and equate it to 0. But a line with slope 0 is a horizontal line. In effect, we are looking for a horizontal tangent of f(x) = 5x-x^2. To give a clearer picture let us look at the graph of f(x) = 5x - x^2.

Figure 4 – Tangent lines of 5x – x2.

From the graph it is clear that the maximum point of the function is where the tangent line (red line) horizontal. In fact, there are only three possible cases that tangent line could be horizontal as shown in Figure 5: first, the minimum of a function (blue graph); second, the inflection point (red graph); and the third is the maximum of the function (green graph).

It should also be noteworthy to say that all the ordered pairs (length, area) or(width, area) in Figure 1 will be on the blue curve in Figure 4.

Figure 5 – Cases of a graph where the tangent is horizontal.

The derivative has many applications and it is seen in many topics in calculus.  In the next Derivative Tutorial, we are going to discuss how the derivative is used in other context.

Summary

  • The derivative of a function is the slope of the line tangent to a function at a particular point.
  • The horizontal line has slope zero.
  • In solving maxima and minima problems, we get the derivative of a function and equate to zero to get the minimum or maximum. We do this because geometrically, we want to get the line tangent to a function at a particular point that is horizontal.
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Derivative in Real Life Context

Calculus and Analysis, College Mathematics

Note: This article is the second part of the derivative concept series. The first part is  The Algebraic and Geometric Meaning of Derivative and the third part is Derivative and the Maximum Area Problem.

If we change the labels of the Figure 2 in the The Algebraic and Geometric Meaning of Derivative article – its x-axis to time and the y-axis to distance – the graph of the secant line is the difference in distance y_2 - y_1 =  100 km – 50 km over the difference  in timex_2 - x_1 = 10:00 – 8:00.  This is shown in the figure below.  This is equivalent to 50 km/2 hrs or 25 km/hr. From the computation above, it is clear that the interpretation of the slope of the secant line is the total distance over the total time or the average speed.

On the other hand, the slope of the tangent line is the speed of the bicycle at exactly 4 o’clock. At exactly 4:00 o’clock the bicycle was traveling 50 km per hour, a lot faster than its average speed. Now, this is reasonable because in real life, the speed of travel is not always constant.  The slope of the tangent line or the speed of travel at a particular point is called the instantaneous speed.

We have also observed from the previous article that as Q approaches P, the secant line’s approximation of the tangent line becomes better and better. This means that as Q approaches P, the average speed becomes closer and closer to the instantaneous speed at P.

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