This is the last page of the Calculus textbook for High School students. I am including the university or college textbook along with the High School textbook. I am going to combine both texts into one textbook, so the student can read the finest and most simple explanation of Calculus, or go to the more elaborate and detailed explanation.
A major problem of education of the sciences, not only in the USA but around the world, is that the textbooks and teachers are incapable of communicating to students on their level of understanding or comprehension. It is because many teachers of science do not know how to teach, and are not required to take training on how to teach. And this problem is intensified in College where the professor usually has no requirement of teaching abilities. If the textbook is well written, then students do not really need the professor teacher, but if the textbook is bad and the teacher is bad, then the student is really hurting. And I have been in many of those situations. I was too young then to realize those situations and ended up with many a "c" grade, whereas the text and teacher deserved a "d or f" grade.
So this 10 page text for High School students is likely to be the finest text on Calculus ever written, because even someone who hates mathematics but wants to know what Calculus is, can read these 10 pages and get a basic understanding of Calculus.
I have given this 10 page text my devoted attention to simplifying the subject. Some may have read the textbook MATHEMATICS, A Human Endeavor, by Harold Jacobs, 1970, as a book on math where no teacher is required because the text is so excellent. But excellent textbooks in science are rare.
This last page, I want to discuss axioms of mathematics and the Maxwell Equations as being the ultimate axioms.
This last page will discuss the idea that all of mathematics comes from the 4 Maxwell Equations which are equations of Calculus.
The 4 Maxwell Equations are too difficult for High School students to learn and they can put that off until they go to College, although I see some High School physics textbooks do cover the 4 Maxwell Equations. My own experience was that the PSSC textbook in High School in the late 1960s covered the Maxwell theory of electricity and magnetism of the 4 laws and we saw demonstrations and films on the 4 laws. But luckily we were not subjected to the mathematics of the Maxwell Equations, because really, few of us in the classroom had even taken Precalculus, and to be forced to do div and curl of the Maxwell Equations would have been rather another pathetic incident of the education system gone amok.
The PSSC physics text was a new approach to teaching physics in High School. I was more of a critic of the way it was taught, since often, the teacher and students "did not know what they were doing or what was going on" and especially during laboratory time. But it did make us aware of what to expect of physics before we reached University. The book did cover the subject of the Maxwell laws in ample detail. But honestly, I think a student's first exposure to physics or chemistry should be a history book of the subject which is pared to the minimum of mathematics. So often our science teaching drowns the students of math and they miss the important ideas, whereas the math should only come in to clarify the ideas. Many a teacher and professor is mistaken when they think the science is the math. Closer to the truth is that the science is a geometrical event such as the demonstrations of the Maxwell laws and the mathematics is there only to detail those demonstrations. It is far better for students to know the Maxwell Equations as the geometrical events -- bar magnet thrust through a closed loop produces an electric current, then to rattle off some mathematical equation.
For those that are going to College and taking physics in College, here are those 4 Maxwell Equations to give you a sneek preview of what you can look forward to, or, for some --to avoid physics:
div*E = r_E charge and the electric field (Coulomb's law)??
div*B = r_B magnetic monopoles and magnetic field (Gauss's law)
??- curlxE = dB + J_B changing magnetic field produces an electric current (Faraday law)
??curlxB = dE + J_E changing electric field or a current produces a magnetic field (Ampere/Maxwell law)
But to the High School students, they still can learn the demonstrations of the 4 laws of the Maxwell Equations. The demonstrations such as Faraday law where a moving bar magnet, thrust into a closed loop wire causes a electric current to arise and flow in the wire. So if the student learns the demonstrations of the Maxwell Equations, that is just as good if not better than learning the equations themselves.
Because the demonstrations, are the equations in geometry math, rather than in quantitative math. So by all means, learn and do the demonstrations of the Maxwell theory laws.
Now the High School student knows what axioms are, because they took Plane Geometry and did some proofs of Plane Geometry where you had to use axioms to build the proof. Remember all those proofs that two triangles are congruent via side angle side or angle side angle.
Now the Maxwell Equations serve as the axioms over all of physics. What that means is that all of physics can be derived by using the Maxwell Equations and since Physics includes all of mathematics, means that all of mathematics comes from just those 4 Maxwell Equations as axioms.
But the Maxwell Equations are too complicated for High School students who must wait until they are in College doing the Maxwell Equations.
However, High School students are ready, equipped and able to do some geometry axioms which are of vital importance to the True Calculus.
So in this last page let me discuss something that all High School students learned in mathematics of axioms of math while in Euclidean Plane Geometry class.
In the axioms of geometry, and all other axiom sets, Old Math had that a point has no length, no width, no depth, yet they also had that a line has no width, no depth, but does have length.
Does the High School student see something wrong with those two axioms of the point and line? Do you sense something not correct, something contradictory? If a point has no length, no width, no depth and since a line is composed of only points, yes, only points, then a line should not have length either. Do you see the contradiction?
Do you see and realize that those axioms of line and point were contradictory, for you cannot have a line with length composed of points with no length.
To escape that logical contradiction, you must impose the idea we learned in this 10 page text on Calculus. We learned that a point in geometry has empty space between it and its successive neighboring point, so that in 10 Grid, the point 0 has empty space before it comes to its first neighbor of 0.1 and it in turn has empty space until it comes to its successive neighbor point of 0.2.
So to solve the contradiction of Euclidean Plane Geometry, that a line or line segment has length but no width and no depth, is solved because the length of a line is produced by all those empty spaces added up. That length comes about by the concept of empty space between points, so that a line is composed of not just points but of points with empty space between successive points.
So in New Math, we need not just point and line but we need empty space. We need these three axioms:
And we see that the empty space not only gives lines their length, but also gives geometry figures such as picketfence their area.
So, we learned well in this 10 page textbook, so well that we even corrected our old Euclidean Plane Geometry textbooks.