笔记—机器学习中的数学基础(2)

全微分

total differential

 

 

 

 

 

 

 

 

 

※ Envisioning total derivatives of a scalar function f(x,y)

 

梯度、方向导数和全微分的关系

梯度、方向导数是向量; 全微分是个点积, 即梯度点积微分向量.

• 概念

gradient: n. 坡度；斜度

a slope or a degree of slope, especially in a road or railway

Physics The rate at which a physical quantity, such as temperature or pressure, changes relative to change in a given variable, especially distance.

梯度：一个物理量，如温度或压力，相对于另一给定可变量（尤指距离）的变化的变化率

 

A vector having coordinate components that are the partial derivatives of a function with respect to its variables.

 

slope 斜率：一个矢量，其坐标分量是一个函数对其变量的偏微分

 

不定积分

原函数的概念:

 

定积分的概念:

 

不定积分的英文解释:

If f(x) has original functions, then all of its original functions {F(x) | F'(x) = f(x)} are defined as the indefinite integral of f(x).

The indefinite integral of f(x), denoted ∫f(x) dx, is defined to be the antiderivative of f(x). In other words, the derivative of ∫f(x) dx is f(x).

※ indefinite integral (不定积分) = antiderivative, primitive integral

 

 

基本积分公式表

 

不定积分的四则运算

 

 

 

定积分

Given a function  that is continuous on the interval [a,b] we divide the interval into n subintervals of equal width, , and from each interval choose a point, .  Then the definite integral of f(x) from a to b is

 

The result of performing the integral is a number that represents the area under the curve of ƒ(x) between the limits and the x -axis.

 

 

Fundamental theorem of calculus (Newton-Leibniz theorem)

※ If f is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of f is known, the definite integral of over that interval is given by

 

 

积分的两种理解

1. 导函数的曲线下被无限分割的矩形面积;
2. 原函数被无限分割的曲线的和. (一个被无限分割的矩形的面积对应一条被无限分割的曲线的长度)