Functions and the Idea of Calculus
Calculus studies how things change and how things accumulate. Before we can speak of change, we need the object that changes: the . This chapter sets up functions and the handful of ``essential'' ones that appear everywh
阅读文章 →Limits and Continuity
The limit is the idea that makes calculus rigorous. ``Instantaneous'' rate of change and ``infinitely thin'' rectangles both hide a limit: a quantity we approach but may never reach. This chapter builds limits intuitivel
阅读文章 →The Derivative
The derivative is the central object of differential calculus and the single most important concept for machine learning. It answers the tangent problem: the instantaneous rate at which a function changes. Once we have i
阅读文章 →Differentiation Rules
Computing every derivative from the limit definition would be exhausting. Fortunately, a small set of rules lets us differentiate almost any function mechanically. The most important of these --- the --- is the mathemati
阅读文章 →Applications of the Derivative
Now that we can compute derivatives, we put them to work. Derivatives find the highest and lowest points of a function (), approximate complicated functions by simple ones ( and Newton's method), and resolve stubborn lim
阅读文章 →The Integral and the Fundamental Theorem
Integration is the second pillar of calculus: the mathematics of . Where the derivative chops a curve into instantaneous rates, the integral sums infinitely many tiny pieces into a total --- an area, a distance, a probab
阅读文章 →Techniques of Integration
Differentiation is mechanical; integration is an art. There is no universal algorithm, so we build a toolkit of techniques --- each one reversing a differentiation rule --- plus numerical methods for the (many) integrals
阅读文章 →Applications of Integration
Integration is far more than ``area under a curve.'' Any quantity built by accumulating infinitely many infinitesimal pieces is an integral: areas between curves, volumes of solids, lengths of arcs, averages, and --- mos
阅读文章 →Sequences, Series, and Taylor Series
Can you add up infinitely many numbers and get a finite total? Sometimes --- and when you can, the result is one of the most powerful tools in mathematics: the ability to represent complicated functions as infinite polyn
阅读文章 →Multivariable Calculus: Partial Derivatives and the Gradient
Machine-learning models depend on millions of parameters at once, so single-variable calculus is not enough. This chapter extends differentiation to functions of several variables. The star object is the --- the vector o
阅读文章 →Multiple Integrals and a Glimpse of Vector Calculus
Just as differentiation extended to many variables, so does integration. accumulate a quantity over a region of two, three, or more dimensions --- the natural setting for probabilities over several variables. The formula
阅读文章 →Optimization and Differential Equations
This chapter gathers the calculus that most directly becomes machine learning: how to find minima of functions of many variables (with and without constraints), the gradient-descent dynamics that training uses, and the d
阅读文章 →Calculus in Machine Learning, Deep Learning, and AI
Here is the payoff. Linear algebra gives AI its --- vectors, matrices, transformations. Calculus gives it the ability to . At the heart of almost every modern machine-learning system is one computational pattern: compute
阅读文章 →Preface: how to use this book
If linear algebra is the language of machine learning, calculus is its . Calculus is the mathematics of things that vary --- how a quantity responds to a small nudge in another (the derivative), and how countless tiny pi
阅读文章 →Notation and Formula Reference
A compact reference for the symbols and identities used throughout the book.
阅读文章 →Vectors and the Geometry of Space
Everything in linear algebra is built from one object: the . Before we can multiply matrices or diagonalize transformations, we need to be completely comfortable with what a vector is, how to combine vectors, and how to
阅读文章 →Systems of Linear Equations
A asks: which values of the unknowns satisfy several linear constraints at once? This is the oldest problem in the subject and still its computational core --- training a linear model, balancing a chemical equation, and
阅读文章 →Matrices and Matrix Algebra
A matrix is the workhorse object of linear algebra. This chapter develops the algebra of matrices --- how to add, multiply, transpose, and invert them --- always keeping sight of the three views from the preface: a matri
阅读文章 →Determinants
The determinant compresses an entire square matrix into a single number that answers two questions at once: and It is less central to computation than students often think (we rarely compute large determinants), but its
阅读文章 →Vector Spaces and the Four Fundamental Subspaces
This is the conceptual heart of linear algebra. Everything so far --- vectors, systems, matrices --- now gets organized by a single powerful abstraction: the and its . The payoff is the , which describes the four subspac
阅读文章 →Linear Transformations and Change of Basis
We have treated matrices as arrays of numbers. This chapter elevates them to their true identity: matrices linear transformations once a basis is fixed. This view explains why matrix multiplication is defined the way it
阅读文章 →Orthogonality, Projections, and Least Squares
Orthogonality is where linear algebra becomes a tool for data. When an exact solution to = does not exist --- the everyday situation with real, noisy, over-determined data --- orthogonality tells us how to find the appro
阅读文章 →Eigenvalues and Eigenvectors
Eigenvalues reveal the hidden axes of a transformation --- the special directions a matrix merely stretches without rotating. They explain the long-term behavior of dynamical systems, the principal directions of data, th
阅读文章 →Symmetric Matrices, the Spectral Theorem, and Quadratic Forms
Symmetric matrices (=) are the best-behaved objects in linear algebra, and --- not coincidentally --- the ones that appear most in machine learning: covariance matrices, Gram matrices, kernel matrices, and Hessians are a
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