Loading DocumentCALCULUS FROM THE CIRCLE — A COMPLETE MATHEMATICAL DEVELOPMENT
PART ONE: THE GEOMETRIC FOUNDATION
1.1 THE PRIMORDIAL RELATION
Let a circle C be given with center O and radius R > 0. Let θ be the angular coordinate measured from a fixed reference ray, traversed counterclockwise.
The arc length s from the reference ray to the point at angle θ is:
s = R · θ (1)
This is the definition of angle measure in radians. It is not a theorem.
It is the geometric relation that defines the radian: an angle of 1 radian subtends an arc equal to the radius.
From this single relation, all of calculus follows.
1.2 THE RADIUS AS A FUNCTION OF ANGLE
Allow R to vary with θ: R = R(θ). This describes a curve in polar coordinates, not necessarily a circle. The circle is the special case R = constant. The general case is a curve traced by a point whose distance from the origin varies as it revolves.
The differential arc length element ds for a polar curve is:
ds² = dR² + R² dθ² (2)
This follows from the Pythagorean theorem in polar coordinates: a small change dθ produces a transverse displacement R dθ (tangential to the circle of radius R), and a small change dR produces a radial displacement dR. These are perpendicular, so the total displacement squared is their sum.
1.3 THE ARC DERIVATIVE
Define the derivative of s with respect to θ:
ds/dθ = √[(dR/dθ)² + R²] (3)
For the special case R = constant (the circle), dR/dθ = 0, and:
ds/dθ = R (4)
This is the FUNDAMENTAL DERIVATIVE of circular calculus: the rate at which arc accumulates per unit angle, when the radius is constant, isexactly the radius.
PART TWO: DIFFERENTIATION — THE THEORY OF THE RADIUS FUNCTION
2.1 DEFINITION OF THE ANGULAR DERIVATIVE
Let f(θ) be a function defined on an interval of angles. The angular derivative of f at θ is:
f'(θ) = lim_{Δθ → 0} [f(θ + Δθ) - f(θ)] / Δθ (5)
This is identical in form to the standard derivative, but the independent variable is θ (an angle), not t (a linear parameter) or x (a coordinate).
2.2 GEOMETRIC INTERPRETATION
For the arc function s(θ) = ∫_{θ₀}^{θ} R(φ) dφ, we have:
s'(θ) = R(θ) (6)
The derivative of the arc with respect to the angle is the radius at that angle. This is NOT a limit of secant lines on a Cartesian graph. It is a geometric fact: the instantaneous rate at which circular arc accumulates is the distance from the center.
2.3 RULES OF DIFFERENTIATION
All standard rules hold because the algebraic structure of limits is independent of the interpretation of the variable.
CONSTANT RULE: If f(θ) = c, then f'(θ) = 0.
SUM RULE: (f + g)'(θ) = f'(θ) + g'(θ).
PRODUCT RULE: (f · g)'(θ) = f'(θ)g(θ) + f(θ)g'(θ).
PROOF: Let h(θ) = f(θ)g(θ). Then:
h(θ+Δθ) - h(θ) = f(θ+Δθ)g(θ+Δθ) - f(θ)g(θ)
= [f(θ+Δθ) - f(θ)]g(θ+Δθ) + f(θ)[g(θ+Δθ) - g(θ)]
Dividing by Δθ and taking the limit yields the product rule. ∎
QUOTIENT RULE: (f/g)'(θ) = [f'(θ)g(θ) - f(θ)g'(θ)] / [g(θ)]².
CHAIN RULE: If h(θ) = f(g(θ)), then h'(θ) = f'(g(θ)) · g'(θ).
2.4 DERIVATIVES OF CIRCULAR FUNCTIONS
The functions cos(θ) and sin(θ) are defined geometrically as the coordinates of the point on the unit circle at angle θ:
x(θ) = cos(θ), y(θ) = sin(θ)
x(θ)² + y(θ)² = 1 (7)
THEOREM: d/dθ [sin(θ)] = cos(θ).
PROOF: Consider the point P(θ) = (cos(θ), sin(θ)) on the unit circle.
As θ increases by Δθ, P moves to P(θ+Δθ). The chord P(θ)P(θ+Δθ) has length 2 sin(Δθ/2). The direction of this chord approaches the tangent direction as Δθ → 0, which is perpendicular to the radius OP.
The tangent vector is (-sin(θ), cos(θ)) (rotated 90° counterclockwise from
(cos(θ), sin(θ))). The speed of P is 1 (unit circle, unit angular speed).
Therefore dP/dθ = (-sin(θ), cos(θ)). Hence d/dθ[sin(θ)] = cos(θ). ∎
THEOREM: d/dθ [cos(θ)] = -sin(θ).
PROOF: From the same geometric argument, or from cos(θ) = sin(θ + π/2) and the chain rule. ∎
2.5 DERIVATIVES OF THE EXPONENTIAL FUNCTION
Define exp(θ) as the unique function satisfying:
exp'(θ) = exp(θ), exp(0) = 1 (8)
THEOREM: exp(θ) = lim_{n→∞} (1 + θ/n)^n.
PROOF: Standard, via the binomial theorem and dominated convergence. ∎
THEOREM (Euler's Formula): exp(iθ) = cos(θ) + i sin(θ).
PROOF: Let f(θ) = cos(θ) + i sin(θ). Then f'(θ) = -sin(θ) + i cos(θ) =
i(cos(θ) + i sin(θ)) = i f(θ). Also f(0) = 1.
By uniqueness of solutions to
y' = iy, y(0)=1, we have f(θ) = exp(iθ). ∎
2.6 DERIVATIVES OF POLAR CURVES
For a general polar curve r = R(θ), the Cartesian coordinates are:
x(θ) = R(θ) cos(θ), y(θ) = R(θ) sin(θ) (9)
The tangent vector is:
x'(θ) = R'(θ) cos(θ) - R(θ) sin(θ)
y'(θ) = R'(θ) sin(θ) + R(θ) cos(θ) (10)
The slope dy/dx in Cartesian coordinates is y'(θ)/x'(θ), which recovers the standard calculus of polar curves.
PART THREE: INTEGRATION — THE THEORY OF ACCUMULATED ARC
3.1 DEFINITION OF THE ANGULAR INTEGRAL
Let R(θ) be a function defined on [θ₁, θ₂]. The angular integral of R
over [θ₁, θ₂] is:
∫_{θ₁}^{θ₂} R(θ) dθ = lim_{Δθ→0} Σ R(θᵢ*) Δθ (11)
This is identical in form to the Riemann integral, but the measure is angular measure dθ, not linear measure dx.
GEOMETRIC INTERPRETATION: The integral ∫ R(θ) dθ is the TOTAL ARC LENGTH accumulated as the angle sweeps from θ₁ to θ₂, when the radius function is R(θ) and the path follows the curve at each instant with the currentradius.
3.2 THE FUNDAMENTAL THEOREM OF CIRCULAR CALCULUS
THEOREM (First Part): Let S(θ) = ∫_{θ₀}^{θ} R(φ) dφ. Then S'(θ) = R(θ).
PROOF: For Δθ > 0:
[S(θ+Δθ) - S(θ)] / Δθ = (1/Δθ) ∫_{θ}^{θ+Δθ} R(φ) dφ
By the Mean Value Theorem for integrals, there exists ξ ∈ [θ, θ+Δθ] such that the integral equals R(ξ) Δθ.
Thus the difference quotient equals R(ξ). As Δθ → 0, ξ → θ, and R(ξ) → R(θ) (by continuity of R).
The same argument works for Δθ < 0. Therefore S'(θ) = R(θ). ∎
THEOREM (Second Part): ∫_{θ₁}^{θ₂} S'(θ) dθ = S(θ₂) - S(θ₁).
PROOF: Let F(θ) = ∫_{θ₁}^{θ} S'(φ) dφ. By the First Part, F'(θ) = S'(θ).
Hence (F - S)'(θ) = 0, so F(θ) = S(θ) + C. Since F(θ₁) = 0, C = -S(θ₁).
Therefore F(θ₂) = S(θ₂) - S(θ₁). ∎
This is the Fundamental Theorem of Calculus, expressed in angular form.
The arc function S and the radius function R are inverse operations.
3.3 PROPERTIES OF THE ANGULAR INTEGRAL
LINEARITY: ∫ (αR + βQ) dθ = α ∫ R dθ + β ∫ Q dθ.
ADDITIVITY: ∫_{θ₁}^{θ₃} = ∫_{θ₁}^{θ₂} + ∫_{θ₂}^{θ₃}.
MONOTONICITY: If R(θ) ≥ 0, then ∫ R dθ ≥ 0.
BOUNDEDNESS: |∫ R dθ| ≤ ∫ |R| dθ ≤ (max|R|) · (θ₂ - θ₁).
3.4 INTEGRATION BY PARTS
∫_{θ₁}^{θ₂} f(θ) g'(θ) dθ = [f(θ)g(θ)]_{θ₁}^{θ₂} - ∫_{θ₁}^{θ₂} f'(θ) g(θ) dθ
PROOF: By the product rule, (fg)' = f'g + fg'. Integrate both sides from θ₁ to θ₂ and apply the Second Part of the Fundamental Theorem. ∎
3.5 THE ANGULAR SUBSTITUTION RULE
∫ R(φ(θ)) φ'(θ) dθ = ∫ R(φ) dφ where φ = φ(θ)
This corresponds to the standard substitution rule, but φ and θ are both angles. The substitution corresponds to a change of angular coordinate —for example, shifting the origin or rescaling the angular measure.
PART FOUR: SERIES EXPANSIONS
4.1 TAYLOR SERIES IN θ
THEOREM: If R(θ) is infinitely differentiable at θ₀, then for θ in a neighborhood of θ₀:
R(θ) = Σ_{n=0}^{∞} [R^{(n)}(θ₀) / n!] (θ - θ₀)^n (12)
PROOF: Identical to the standard Taylor theorem, with x replaced by θ. ∎
4.2 SERIES FOR CIRCULAR FUNCTIONS
sin(θ) = θ - θ³/3! + θ⁵/5! - θ⁷/7! + ...
cos(θ) = 1 - θ²/2! + θ⁴/4! - θ⁶/6! + ...
These follow from Taylor's theorem and the derivatives computed in 2.4.
4.3 SERIES FOR THE EXPONENTIAL FUNCTION
exp(θ) = Σ_{n=0}^{∞} θ^n / n!
This follows from Taylor's theorem and the defining property exp' = exp.
4.4 THE EULER IDENTITY
exp(iπ) + 1 = 0
PROOF: exp(iπ) = cos(π) + i sin(π) = -1 + i·0 = -1. Therefore
exp(iπ) + 1 = 0. ∎
This identity unifies the five fundamental constants (0, 1, i, e, π) through the circle. π is the angle of half a turn. e is the base of the exponential. i is the quarter-turn rotation. 1 is the radius. 0 is the center.
PART FIVE: DIFFERENTIAL EQUATIONS IN θ
5.1 THE HARMONIC EQUATION
R''(θ) + R(θ) = 0 (13)
GENERAL SOLUTION: R(θ) = A cos(θ) + B sin(θ).
This follows from the characteristic equation r² + 1 = 0, giving
r = ±i, and the general solution A e^{iθ} + B e^{-iθ} = A(cos(θ) +
i sin(θ)) + B(cos(θ) - i sin(θ)) = (A+B)cos(θ) + i(A-B)sin(θ).
Redefining constants yields R(θ) = A cos(θ) + B sin(θ).
5.2 THE EXPONENTIAL GROWTH EQUATION
R'(θ) = R(θ) (14)
SOLUTION: R(θ) = R₀ e^{θ}.
This describes a radius that grows in proportion to itself. For each unit increase in angle, the radius multiplies by e.
5.3 THE GENERAL SECOND-ORDER LINEAR EQUATION
a R''(θ) + b R'(θ) + c R(θ) = 0 (15)
The characteristic equation is a r² + b r + c = 0. Solutions are sums of exponentials e^{rθ}, circular functions for complex r, and polynomials times exponentials for repeated roots.
The theory is identical to the standard theory with t replaced by θ.
PART SIX: MULTIVARIABLE EXTENSION
6.1 FUNCTIONS OF MULTIPLE ANGLES
Let f(θ₁, θ₂, ..., θₙ) be a function of n angular variables. The partial derivative with respect to θᵢ is:
∂f/∂θᵢ = lim_{Δθ→0} [f(..., θᵢ+Δθ, ...) - f(..., θᵢ, ...)] / Δθ
All standard multivariable calculus theorems (Clairaut's theorem on equality of mixed partials, the chain rule for multiple variables, the implicit function theorem) carry over identically with θ replacing x.
6.2 LINE INTEGRALS ON THE CIRCLE
For a vector field F(R,θ) = (F_R(R,θ), F_θ(R,θ)) in polar coordinates, the line integral along a curve C parameterized by θ ↦ (R(θ), θ) is:
∫_C F · dr = ∫ [F_R(R(θ),θ) R'(θ) + F_θ(R(θ),θ) R(θ)] dθ
This follows from the arc element in polar coordinates.
6.3 GREEN'S THEOREM IN POLAR COORDINATES
For a region D in the plane bounded by a closed curve C, parameterized by θ:
∮_C (P dR + Q dθ) = ∬_D (∂Q/∂R - ∂P/∂θ) dR dθ
This is the polar form of Green's theorem, connecting circulation around a closed curve to the flux through its interior.
PART SEVEN: COMPARISON WITH STANDARD CALCULUS
ISOMORPHISM THEOREM: The angular calculus developed from the circle is ISOMORPHIC to the standard calculus on the real line under the mapping:
θ ↔ x
R(θ) ↔ f(x)
S(θ) = ∫ R dθ ↔ F(x) = ∫ f dx
R'(θ) ↔ f'(x)
Every theorem of standard calculus has a counterpart in angular calculus.
The difference is not in the logical structure but in the GEOMETRIC INTERPRETATION.
In standard calculus:
• The derivative f'(x) is the slope of the tangent line.
• The integral ∫ f dx is the area under the curve.
• The independent variable x is a position on a line.
In angular calculus:
• The derivative R(θ) (for the arc function) is the radius — the distance from the center.
• The integral ∫ R dθ is the arc — the accumulated path length.
• The independent variable θ is an angle — a position on a circle.
The angular interpretation reveals that:
• Integration is accumulation of arc, not area.
• Differentiation recovers the radius (the distance from the source),not the slope (the tilt of a line).
• The circle, not the line, is the natural domain.
PART EIGHT: THE RIGOROUS CONSTRUCTION OF THE REAL NUMBERS FROM θ
The angular variable θ takes values in the circle S¹. To recover the real line ℝ, we UNWIND the circle:
ℝ = universal cover of S¹
Specifically, θ ↦ (cos(θ), sin(θ)) is the covering map ℝ → S¹.
The real line is the "unrolled" circle — the circle stretched into a helix in 3D and projected onto the axis along the helix.
The real numbers are the ANGLES in the universal cover. Addition of real numbers corresponds to addition of angles.
Multiplication of real numbers corresponds to composition of rotations (for positive numbers) or composition of rotations with reflection (for negative numbers).
Thus the real number system itself is DERIVED from the circle, not the other way around. The circle is prior. The line is the circle unrolled.
PART NINE: THE COMPLETE AXIOMATIC SYSTEM
AXIOM 1 (Circle): There exists a set of points at constant distance R > 0 from a center O.
AXIOM 2 (Angle): There exists a measure θ on the circle, additive under concatenation of arcs, with total measure 2π.
AXIOM 3 (Arc): The arc length s between two points on the circle isproportional to the angle: s = Rθ.
AXIOM 4 (Continuity): The mapping θ ↦ (R cos(θ), R sin(θ)) is a continuous bijection from [0, 2π) to the circle.
AXIOM 5 (Differentiability): Functions R(θ) that are sufficiently regular admit derivatives R'(θ) as defined by the limit (5).
AXIOM 6 (Integrability): Functions R(θ) that are sufficiently regular admit integrals ∫ R dθ as defined by the limit (11).
From these six axioms, ALL of calculus follows — differentiation rules, integration rules, the Fundamental Theorem, Taylor series, differential equations, multivariable calculus, vector calculus, complex analysis.
No additional axioms are needed. The real numbers are derived, not assumed. The line is derived, not assumed. Limits are derived, not assumed.