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calculus

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Jul 9, 2026 · Artificial Intelligence

The Three Math Pillars Powering Modern Large Language Models

Beyond compute, data, and Transformer architecture, large language models rely on three core mathematical disciplines—linear algebra for representations, probability and statistics for modeling, and calculus for optimization—each of which underpins token embeddings, attention mechanisms, training objectives, and scaling laws.

Attention MechanismLarge Language Modelscalculus
0 likes · 11 min read
The Three Math Pillars Powering Modern Large Language Models
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Sep 27, 2025 · Fundamentals

What Math Lies Hidden in Tang Poetry? A Modeling Exploration

This article examines classic Tang poems through modern mathematical modeling, revealing how geometric optics, probability, calculus, information theory, and network analysis can be applied to literary works to uncover the underlying mathematical structures and interdisciplinary insights.

GeometryTang Poetrycalculus
0 likes · 10 min read
What Math Lies Hidden in Tang Poetry? A Modeling Exploration
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Jun 12, 2025 · Fundamentals

Why Did Newton Call Calculus ‘God’s Language’? The Story Behind the Equation

From a wartime interview where Feynman urged a novelist to learn calculus as the language of God, through Newton's plague‑ridden youth and his hidden use of infinitesimals, to the Leibniz‑Newton controversy and modern applications, this article traces how calculus became the indispensable tool of science and engineering.

MathematicsNewtonPhysics
0 likes · 22 min read
Why Did Newton Call Calculus ‘God’s Language’? The Story Behind the Equation
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Dec 19, 2024 · Fundamentals

How Calculus Powers Optimal Investment Strategies

This article explains how calculus and mathematical modeling can maximize investment returns, balance risk and reward, and optimize portfolio allocation, using derivative analysis, utility functions, Lagrange multipliers, and integration for long‑term planning, illustrating the scientific power behind financial decision‑making.

calculusinvestmentmathematical modeling
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How Calculus Powers Optimal Investment Strategies
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Aug 28, 2024 · Fundamentals

Why Calculus Matters: From Static Geometry to Dynamic Change

Calculus, the dynamic mathematics of infinitesimals, emerged to solve problems like instantaneous velocity, tangent slopes, and areas under curves, transforming static algebra and geometry into tools for analyzing continuous change, as illustrated by examples of expanding squares, sphere volumes, and tidal movements.

DerivativesLimitscalculus
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Why Calculus Matters: From Static Geometry to Dynamic Change
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Dec 28, 2023 · Fundamentals

How Modern Mathematics Solves Zeno’s Classic Paradoxes

This article explains Zeno’s four classic paradoxes—Achilles and the Tortoise, the Dichotomy, the Arrow, and the Stadium—and shows how modern mathematics, especially infinite series and calculus, resolves each paradox, illustrating the bridge between ancient philosophy and contemporary mathematical thought.

Zenocalculusinfinite series
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How Modern Mathematics Solves Zeno’s Classic Paradoxes
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Oct 3, 2023 · Fundamentals

Infinite Series Explained: Concepts, Convergence Tests, Real-World Uses

This article introduces infinite series, outlines common types such as arithmetic, geometric, harmonic, p‑series, Taylor, and Fourier series, explains key convergence tests, and demonstrates practical applications in physics, engineering, economics, and mathematical modeling.

ApplicationsFourier SeriesTaylor series
0 likes · 8 min read
Infinite Series Explained: Concepts, Convergence Tests, Real-World Uses
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Sep 2, 2023 · Fundamentals

How Calculus Evolved: From Ancient Approximations to Modern Mathematics

Calculus, the study of change and limits, has evolved over millennia—from ancient Greek approximations by Archimedes, through medieval contributions in the Islamic world, to the independent breakthroughs of Newton and Leibniz in the 17th century, and later formalizations by Cauchy and others that shaped modern science.

CauchyLeibnizMathematics History
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How Calculus Evolved: From Ancient Approximations to Modern Mathematics
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Feb 23, 2023 · Fundamentals

Why Derivatives Matter: From Motion to Modern Science

Derivatives, the core concept of calculus, originated from Newton’s instantaneous speed, Leibniz’s tangent, and Fermat’s extremum problems, and are defined as the limit of a function’s change ratio, linking continuity, geometry, and physics, while finding extensive applications across science, economics, and biology.

ApplicationsMathematicsPhysics
0 likes · 10 min read
Why Derivatives Matter: From Motion to Modern Science
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Sep 26, 2022 · Fundamentals

Deriving Scalars with Respect to Vectors Using the Definition Method

This article explains how to compute the derivative of a scalar-valued function with respect to a vector by applying the definition method, illustrating the process with simple and more complex examples, outlining basic differentiation rules, and discussing the limitations of the definition approach for vector‑matrix derivatives.

calculusdefinition methodderivative
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Deriving Scalars with Respect to Vectors Using the Definition Method
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Aug 11, 2022 · Fundamentals

Understanding Gradients: How Directional Derivatives Reveal Maximum Change

The article explains that the gradient of a function is a vector pointing in the direction of greatest increase, representing the maximal directional derivative, and illustrates this concept for single‑variable and two‑variable functions with diagrams, showing how unit vectors determine rates of change along any direction.

calculusdirectional derivativegradient
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Understanding Gradients: How Directional Derivatives Reveal Maximum Change
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Jul 13, 2022 · Fundamentals

Key Properties of Integrals and Differentials Explained

This article outlines the fundamental properties of integrals—including the sum rule, constant factor rule, interval splitting, and limit‑based computation—and introduces differentials, their basic properties, and how they relate to integral calculus.

AnalysisMathematicscalculus
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Key Properties of Integrals and Differentials Explained
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Jul 10, 2022 · Fundamentals

Definite Integrals Demystified: Tables, Properties, and an Oil Consumption Case

This article presents a concise reference of basic integral tables, explains key properties of definite integrals such as additivity and scalar multiplication, and demonstrates their practical use through a detailed case calculating total oil consumption between 1970 and 1990 via differential methods.

applied mathematicscalculusdefinite integrals
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Definite Integrals Demystified: Tables, Properties, and an Oil Consumption Case
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Jul 9, 2022 · Fundamentals

Unlocking the Power of the Fundamental Theorem of Calculus

This article explains how the Fundamental Theorem of Calculus bridges derivatives and integrals, provides a straightforward method for evaluating integrals via antiderivatives, illustrates the concept with telescopic sums, and walks through several concrete examples to demonstrate its practical use.

DerivativesIntegrationMathematics
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Unlocking the Power of the Fundamental Theorem of Calculus
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May 20, 2022 · Fundamentals

Understanding Derivatives: From Rate of Change to Function Slopes

This article explains the concept of derivatives, covering the rate of change, the limit definition of instantaneous change, geometric interpretation as tangent slopes, and provides derivative formulas and brief proofs for common functions such as constants, linear, quadratic, power, exponential, sine, and cosine.

Functioncalculusderivative
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Understanding Derivatives: From Rate of Change to Function Slopes
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May 20, 2022 · Fundamentals

Master Derivative Rules: Sum, Product, and Chain Rule Made Simple

This article explains how to differentiate sums, differences, and products of functions, introduces function composition, and derives the chain rule, providing proofs and clear examples to help readers master fundamental calculus differentiation techniques.

Derivativescalculuschain rule
0 likes · 2 min read
Master Derivative Rules: Sum, Product, and Chain Rule Made Simple
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May 11, 2022 · Fundamentals

How Derivatives Find Global and Local Optima (Minima & Maxima)

This article explains how to use derivatives to locate a function’s global and local minimum or maximum points, defines critical points, distinguishes between global and local extrema, and demonstrates the process with quadratic and cubic examples illustrated by diagrams.

DerivativesOptimizationcalculus
0 likes · 4 min read
How Derivatives Find Global and Local Optima (Minima & Maxima)
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May 11, 2022 · Fundamentals

What Is a Derivative? Master Rate of Change, Tangent Slopes, and Core Functions

This article explains the concept of derivatives by introducing the rate of change, defining the instantaneous rate as a limit, interpreting derivatives geometrically as tangent slopes, and presenting the derivatives of common functions such as constants, linear, quadratic, power, exponential, sine, and cosine.

calculuscommon functionsderivative
0 likes · 4 min read
What Is a Derivative? Master Rate of Change, Tangent Slopes, and Core Functions
Python Programming Learning Circle
Python Programming Learning Circle
Mar 12, 2020 · Fundamentals

Fundamentals of Derivatives and Partial Derivatives for Neural Networks

This article introduces the mathematical foundations of derivatives and partial derivatives, explains their role in optimizing neural network parameters, covers basic derivative formulas, linear properties, sigmoid derivative, minimum conditions, and constrained optimization using Lagrange multipliers, providing a comprehensive guide for machine‑learning practitioners.

DerivativesOptimizationcalculus
0 likes · 8 min read
Fundamentals of Derivatives and Partial Derivatives for Neural Networks