THE ARK OF MATHEMATICS (5 book series)

The ARK OF MATHEMATICS shows how to prove the fundamentals of Geometry starting from the concrete definitions. The first lesson in this part starts with showing the concrete concept of an angle and arc length using the definition of a circle, its radius, its diameter, and its circumference. The second lesson in this part continues with showing how to prove the fundamental properties of a triangle from Geometry including the sine function, cosine function, Pythagorean Theorem, and angle addition formulas. This part concludes with a summary of Euclid's Axioms and the axioms for Trigonometry used to prove all of Geometry and trigonometric identities.
THE ARK OF MATHEMATICS PART 2: PROVING TRIGONOMETRY, TRIANGLES, AND CIRCLES
The ARK OF MATHEMATICS shows how to prove the fundamentals of Geometry and Trigonometry starting from the concrete definitions using Algebra and Calculus. The first lesson in this part starts with showing how to prove the sine and cosine double angle properties, half angle properties, law of sine, and law of cosine using the axioms for Trigonometry previously proved in Part 1. The first lesson then continues with showing how to prove the fundamental Sine and Cosine Calculus limits using Geometry, which are used to create Calculus Taylor Series representations of the Sine and Cosine functions. The Calculus Taylor Series are used to prove Euler's Formula, which is then used to find the exact values of the Sine and Cosine functions associated with 15 degree increments around the unit circle. The second lesson in this part starts with showing how to prove important trigonometric identities of the Tangent function such as the angle addition formula, double angle formula, and half angle formula, The second lesson in this part continues with showing how to calculate the value of "Pi" (3.14159....) using Geometry and Trigonometric Identities. The second lesson in this part concludes with proving Heron's Formula for calculating the area of a triangle based only on the lengths of its sides, which is then used to derive the formula for the area of a circle using Calculus limits.
THE ARK OF MATHEMATICS PART 3: PROVING VECTORS AND VECTOR PRODUCTS
The ARK OF MATHEMATICS shows how to prove the fundamentals of Vectors for application in Engineering and Physics starting from the concrete definitions in Trigonometry and Geometry. The first lesson starts with understanding the concepts of vector addition, vector subtraction, and scalar multiplication, which are used to calculate the magnitude and components of a vector in 2 dimensions and 3 dimensions. The second lesson starts with the concrete definition of a "dot product" of two vectors using the idea of a projection of one vector onto another vector, which can be either be calculated using the magnitude of the vectors and the angle between or be calculated using the components of both vectors. It is shown how to calculate the dot product between two vectors in both 2 dimensions and 3 dimensions. The dot product of a vector with a unit vector in a specific direction is used to calculate the component of vector in that direction. The second lesson continues with the concrete definition of a "cross product" of two vectors being used to calculate a new vector that has magnitude equal to the area of the parallelogram spanned by the two vectors, which is pointing in a direction perpendicular to that parallelogram. It is shown how to calculate the cross product of two vectors, which can be either be calculated using the components of both vectors or by using the determinant of a square matrix filled with the unit vectors, components of the first vector, and components of the second vector. The second lesson finishes with a proof of the dot product identities, cross product identities, and proof of how to calculate the volume of a parallelepiped (a 3 dimensional generalization of a parallelogram) spanned by three vectors using the scalar triple product.
THE ARK OF MATHEMATICS PART 4: MULTIVARIABLE CALCULUS INTEGRALS
The ARK OF MATHEMATICS shows how to prove the fundamentals of Vectors for application toward Multivariable Calculus Integrals in Engineering and Physics starting from the concrete definitions in Vectors, Geometry. The first lesson starts by showing how to generalize rectangular and polar coordinates in two dimensions into rectangular, polar, and spherical coordinates in three dimensions. The second lesson shows how to calculate the length of a curve in two dimensions using line integrals, which utilizes the Pythagorean Theorem in the length differential. It is shown how to prove the formula for the circumference of a circle. The third lesson shows how to generalize to three dimensions in order to calculate the length of a curve in three dimensions using line integrals, which utilizes a three dimensional generalization of the Pythagorean Theorem in the length differential. It is shown how to calculate a formula for length of a curve that travels along a helix. The fourth lesson shows how to calculate area in two dimensions using surface integrals, which utilizes a cross product in the area differential. It is shown how to calculate a formula for the area of a circle using a surface integral with polar coordinates. The fifth lesson shows how to generalize to three dimensions in order to calculate area in three dimensions using surface integrals, which utilizes a cross product in the area differential. It is shown how to calculate a formula for the surface area of a cylinder and a cone using surface integrals with polar coordinates. It is shown how to calculate the formula for the surface area of a sphere using spherical coordinates. The sixth lesson shows how to calculate volume in three dimensions using volume integrals, which utilizes a scalar triple product in the volume differential. It is shown how to calculate a formula for the volume of a cylinder and cone using cylindrical coordinates. It is shown how to calculate a formula for the volume of a sphere using spherical coordinates.
THE ARK OF MATHEMATICS PART 5: VECTOR CALCULUS FOR ELECTRICITY AND MAGNETISM
The ARK OF MATHEMATICS shows how to prove the fundamentals of Vector Calculus for application toward Advanced Placement high school and college Physics and Engineering involving Electricity and Magnetism. The first lesson generalizes the Chain Rule from Calculus into a multivariable Chain Rule used to define the gradient vector, divergence operator, curl operator, and Laplacian operator. The first lesson continues on how to generalize the Product Rule and Quotient Rule from Calculus to prove calculus properties of the gradient vector, divergence operator, curl operator, and Laplacian operator. The first lesson finishes by showing how to easily integrate the dot product of a vector with one of its odd order derivatives and how to easily integrate the cross product of a vector with one of its even order derivatives. The second lesson starts by showing how to calculate the flux of a vector field through a surface. The flux of the electric field through a closed surface tells how much charge is inside the surface. The divergence theorem (Gauss's Law) is used to calculate the electric field surrounding a spherically symmetric charge distribution or a point charge. The flux of the magnetic field through a closed surface is always zero. The third lessons starts by showing how to calculate the circulation of a vector field around a closed loop. The circulation of the electric field is caused by a changing magnetic field. The circulation of the magnetic field is caused by a moving charge or a changing electric field. The curl theorem (Stokes' Theorem) is used to calculate the magnetic field surrounding a straight wire with a steady current. The lesson finishes with a summary of the Lorentz Force and Maxwell's Equations for Electricity and Magnetism.
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