∇· 散度计算器（输入向量场，二维/三维）Divergence Calculator (Enter a Vector Field, 2D or 3D)
输入向量场 F = (P, Q) 或 (P, Q, R) 的分量表达式（如 x^2*y、sin(z)），在指定点用数值微分求 ∂P/∂x、∂Q/∂y、∂R/∂z 和散度 ∇·F，并说明该点是“源”还是“汇”。也保留由网格上相邻函数值做中心差分估算的方法。本地处理。Enter the components of a vector field F = (P, Q) or (P, Q, R) as expressions (such as x^2*y or sin(z)) to get ∂P/∂x, ∂Q/∂y, ∂R/∂z and the divergence ∇·F at a point by numerical differentiation, and see whether the point is a source or a sink. The central-difference estimate from neighbouring grid values is still available. Processed locally.
使用说明与常见问题How-to & FAQ
「散度计算器（输入向量场，二维/三维）」是什么？What is the Divergence Calculator (Enter a Vector Field, 2D or 3D) tool?
输入向量场 F = (P, Q) 或 (P, Q, R) 的分量表达式（如 x^2*y、sin(z)），在指定点用数值微分求 ∂P/∂x、∂Q/∂y、∂R/∂z 和散度 ∇·F，并说明该点是“源”还是“汇”。也保留由网格上相邻函数值做中心差分估算的方法。本地处理。Enter the components of a vector field F = (P, Q) or (P, Q, R) as expressions (such as x^2*y or sin(z)) to get ∂P/∂x, ∂Q/∂y, ∂R/∂z and the divergence ∇·F at a point by numerical differentiation, and see whether the point is a source or a sink. The central-difference estimate from neighbouring grid values is still available. Processed locally.
如何使用「散度计算器（输入向量场，二维/三维）」？How do I use the Divergence Calculator (Enter a Vector Field, 2D or 3D) tool?
在「散度计算器（输入向量场，二维/三维）」中，填写「方式」、「P(x, y, z)」、「Q(x, y, z)」、「R(x, y, z)（二维留空）」、「点 (x, y, z)」等参数，结果会随输入实时计算并显示在下方。In the Divergence Calculator (Enter a Vector Field, 2D or 3D): Fill in "Method", "P(x, y, z)", "Q(x, y, z)", "R(x, y, z) (blank for 2D)", "Point (x, y, z)"; results are recalculated instantly as you type.
散度怎么算？How is divergence calculated?
∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z，即每个分量对自己方向的变量求偏导后相加。例如 F = (x²y, xy) 时 ∇·F = 2xy + x，在 (1, 2) 处为 5。∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z: each component is differentiated with respect to its own variable and the results are added. For F = (x²y, xy), ∇·F = 2xy + x, which is 5 at (1, 2).
表达式怎么写？How do I write the expressions?
使用 x、y、z 和 + − * / ^，以及 sin、cos、exp、ln、sqrt 等函数，例如 x^2*y、exp(-z)*sin(x)。二维向量场把 R 留空。Use x, y, z with + − * / ^ and functions such as sin, cos, exp, ln and sqrt, for example x^2*y or exp(-z)*sin(x). Leave R blank for a 2D field.
结果准确吗？How accurate is it?
偏导数用中心差分加 Richardson 外推计算，对光滑函数通常准确到 8 位左右；在不连续或不可微的点结果不可靠。Partial derivatives use central differences with Richardson extrapolation, usually good to about 8 digits for smooth functions; results are unreliable at points where the field is not smooth.
能举个计算实例吗？Can you show a worked example?
例如在「散度计算器（输入向量场，二维/三维）」中填 方式 = 输入分量表达式，P(x, y, z) = x^2*y，Q(x, y, z) = x*y，点 (x, y, z) = 1, 2, 0，结果区显示：散度 ∇·F：5；ℹ R 为空，按二维向量场计算，忽略 z 坐标。；∂P/∂x：4；∂Q/∂y：1；F 在该点：(2, 2)；含义：散度为正：该点附近是“源”，向外流出For example, entering Method = Enter component expressions, P(x, y, z) = x^2*y, Q(x, y, z) = x*y, Point (x, y, z) = 1, 2, 0 in the Divergence Calculator (Enter a Vector Field, 2D or 3D) gives: Divergence ∇·F: 5; ℹ R is blank, so the field is treated as 2D and the z coordinate is ignored.; ∂P/∂x: 4; ∂Q/∂y: 1; F at the point: (2, 2); Meaning: Positive: a source, with net outflow near the point
哪些输入会被检查？Which inputs are validated?
「散度计算器（输入向量场，二维/三维）」会检查这些输入——「P(x+h,y)」、「P(x−h,y)」、「Q(x,y+h)」、「Q(x,y−h)」：可留空；「步长 h」：所选方式用到时必须填写，须大于 0；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Divergence Calculator (Enter a Vector Field, 2D or 3D) checks its inputs — "P(x+h,y)", "P(x−h,y)", "Q(x,y+h)", "Q(x,y−h)": optional; "Step h": required when the chosen option uses it, must be greater than 0; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.
有哪些选项可以调整？What options can I adjust?
「散度计算器（输入向量场，二维/三维）」的设置：「方式」可选 输入分量表达式、网格函数值中心差分（默认 输入分量表达式）；「P(x, y, z)」默认 x^2*y；「Q(x, y, z)」默认 x*y；「R(x, y, z)（二维留空）」可留空；「点 (x, y, z)」默认 1, 2, 0；「P(x+h,y)」默认 1.1；「P(x−h,y)」默认 0.9；「Q(x,y+h)」默认 2.2；「Q(x,y−h)」默认 1.8；「步长 h」默认 0.1。每次生成结果都按当前设置处理。Settings in the Divergence Calculator (Enter a Vector Field, 2D or 3D) — "Method": Enter component expressions, Central differences from grid values (default Enter component expressions); "P(x, y, z)" default x^2*y; "Q(x, y, z)" default x*y; "R(x, y, z) (blank for 2D)" optional; "Point (x, y, z)" default 1, 2, 0; "P(x+h,y)" default 1.1; "P(x−h,y)" default 0.9; "Q(x,y+h)" default 2.2; "Q(x,y−h)" default 1.8; "Step h" default 0.1. The result always uses the current settings.
「散度计算器（输入向量场，二维/三维）」免费吗？需要注册吗？Is the Divergence Calculator (Enter a Vector Field, 2D or 3D) free? Do I need an account?
「散度计算器（输入向量场，二维/三维）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Divergence Calculator (Enter a Vector Field, 2D or 3D) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.