非保守力 与 格林公式

CHANGELOG

6.22.2021 开始动笔。
6.24 Add Metor and Introduction.

Metor

The difference between math and physics is that physics describes our universe, while math describes any potential universe. ————Unamed

Introduction

连续性方程或输运方程是描述某种量输运的方程。当应用于守恒量时，它特别简单和强大，但它可以推广应用于任何广泛的量。由于质量、能量、动量、电荷等自然因素 量在各自适当的条件下守恒，各种物理现象可用连续性方程来描述。
连续性方程是守恒定律的一种更强的局部形式。例如，能量守恒定律的一个弱版本指出，能量既不能被创造也不能被毁灭，即宇宙中的能量总量是固定的。这种说法并不排除一种可能性，即一定量的能量可以从一个点消失，同时出现在另一个点上。一个更强有力的说法是能量是局部守恒的：能量既不能被创造也不能被破坏，也不能从一个地方“传送”到另一个地方它只能通过连续的流动来移动。连续性方程就是用数学的方法来表达这种说法的。例如，电荷的连续性方程指出，任何空间体积中的电荷量只能通过通过其边界流入或流出该体积的电流量来改变。
一般来说，连续性方程可以包括“源”和“汇”项，这使得它们能够描述经常但并不总是守恒的量，例如可以通过化学反应产生或破坏的分子物种的密度。在一个日常的例子中，有一个连续性方程来表示活着的人数；它有一个“源项”来解释出生的人，还有一个“汇项”来解释死亡的人。
任何连续性方程都可以表示为适用于任何有限区域的“积分形式”（用通量积分表示），也可以表示为适用于某一点的“微分形式”（用散度算子表示）。
连续性方程是更具体的输运方程的基础，如对流扩散方程、玻耳兹曼输运方程和纳维-斯托克斯方程。
由连续性方程控制的流可以使用Sankey图可视化。

A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.
Continuity equations are a stronger, local form of conservation laws. For example, a weak version of the law of conservation of energy states that energy can neither be created nor destroyed—i.e., the total amount of energy in the universe is fixed. This statement does not rule out the possibility that a quantity of energy could disappear from one point while simultaneously appearing at another point. A stronger statement is that energy is locally conserved: energy can neither be created nor destroyed, nor can it "teleport" from one place to another—it can only move by a continuous flow. A continuity equation is the mathematical way to express this kind of statement. For example, the continuity equation for electric charge states that the amount of electric charge in any volume of space can only change by the amount of electric current flowing into or out of that volume through its boundaries.
Continuity equations more generally can include "source" and "sink" terms, which allow them to describe quantities that are often but not always conserved, such as the density of a molecular species which can be created or destroyed by chemical reactions. In an everyday example, there is a continuity equation for the number of people alive; it has a "source term" to account for people being born, and a "sink term" to account for people dying.
Any continuity equation can be expressed in an "integral form" (in terms of a flux integral), which applies to any finite region, or in a "differential form" (in terms of the divergence operator) which applies at a point.
Continuity equations underlie more specific transport equations such as the convection–diffusion equation, Boltzmann transport equation, and Navier–Stokes equations.
Flows governed by continuity equations can be visualized using a Sankey diagram.