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.
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.
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.
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.
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.
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.
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.
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.
This article explores the origin and meaning of the natural logarithm base e, covering its mathematical definition as a limit, historical naming by Euler and Leibniz, why it simplifies logarithmic derivatives, and a proof of its existence using the monotone bounded theorem.
This article compiles a comprehensive list of previously published WeChat posts from the “Model Perspective” series covering core calculus topics such as single‑variable functions, derivatives, integrals, and multivariable calculus, providing readers with quick access to each detailed tutorial.
This article explains the definition of integrals by illustrating how the area under a curve can be approximated with increasingly fine rectangles, leading to the limit that forms the integral, and connects this geometric intuition to the formal calculus theorem.
This article explains that the gradient of a multivariable function is a vector indicating the direction of greatest increase, shows how directional derivatives are computed, and demonstrates that the maximum rate of change occurs when moving along the gradient direction.
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.
This article aggregates the "Model Perspective" public account's past posts on linear algebra and upcoming calculus summaries, links to comprehensive model‑related articles, and provides a collection of Python NumPy tutorials for applying linear‑algebra concepts in programming.
This article compiles a series of WeChat posts covering calculus fundamentals—including limits, derivatives, integration techniques, gradient descent, and multivariable concepts—providing direct links to each detailed tutorial for learners to explore.
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.
This article explains the integration by parts technique, derived from the product rule, and walks through five detailed examples that show how to rewrite integrals as products of functions, apply the formula, and solve the resulting equations.
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.
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.
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.
This article explains the concept of definite integrals by illustrating how the area under a curve can be approximated with rectangles, leading to the formal definition of integration as the limit of these sums, and connects it to the Fundamental Theorem of Calculus.
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.
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.
This article explains how derivatives can be used to locate global and local minima or maxima, defines critical points—including saddle points—and demonstrates the process with quadratic and cubic function examples, illustrating each concept with clear diagrams.
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.
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.
Explore the core differentiation rules—including the sum, difference, and product rules—followed by functional composition and the chain rule, with clear derivations and proofs that build on basic calculus concepts to help you differentiate complex functions confidently.
This article explains the chain rule for composite functions, covering both single‑variable and multi‑variable scenarios, illustrating how derivatives of nested functions are computed and how the rule applies to common machine‑learning expressions such as the sigmoid of a linear combination.
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.
The article outlines key mathematical areas—probability & statistics, linear algebra, calculus, information theory, and convex optimization—essential for deep learning, and recommends specific lecture notes and textbooks for each topic to guide learners toward effective study.
This lesson deepens students' understanding of function monotonicity by combining graphical and definitional methods to determine extrema and ranges, exploring quadratic and linear cases, parameter sensitivity, and the broader relevance of monotonic concepts in mathematics.
