标量场、向量场与散度Scalar Fields, Vector Fields & Divergence
一个向量场在某点附近是净流出、净流入还是两者平衡？Is a vector field locally flowing out, flowing in, or balanced around a point?
前置知识：Prerequisites: B09：偏导数与梯度；B17：积分直觉。B09: partial derivatives and gradients; B17: integral intuition.
复习前置：Review prerequisites: B09 · 偏导数与梯度Partial Derivatives & Gradients · B17 · 积分直觉与连续分布Integral Intuition & Continuous Distributions
学完后你能：After this lesson you can: 区分场类型，计算源、汇与旋转场的散度和单位面积通量。Distinguish field types and calculate divergence and flux density for source, sink and rotation fields.
动手观察Explore
切换源、汇与旋转场，改变小方框边长，比较净通量与单位面积的散度。Switch between source, sink and rotation fields; change the box side and compare net flux with divergence per unit area.
从观察到解释From observation to explanation
场给每个位置一个值A field assigns a value to each position
标量场在每个位置给一个数，例如温度；向量场给一个带方向的箭头，例如流速。散度讨论的是向量场在局部的净流出。A scalar field assigns a number to each position, such as temperature; a vector field assigns a directional arrow, such as velocity. Divergence concerns local net outflow of a vector field.
小方框的收支Accounting around a small box
对 F=(x,y)，边长为 h 的方框净向外通量是 2h²，除以面积 h² 得散度 2。放大方框时通量和面积都变大，但本例的单位面积净流出不变。For F=(x,y), a square of side h has net outward flux 2h²; dividing by area h² gives divergence 2. Enlarging the box increases flux and area together, while the outflow per unit area stays constant in this example.
源、汇与旋转Sources, sinks and rotation
F=(−x,−y) 的散度为 −2，是汇；F=(−y,x) 的散度为 0，是纯旋转示例。箭头运动明显并不代表净流出非零。F=(−x,−y) has divergence −2 and is a sink; F=(−y,x) has divergence 0 and is a pure rotation example. Visible arrow motion does not imply nonzero net outflow.
公式与符号Formula and symbols
F=(Fₓ,Fᵧ) 是二维向量场；∂Fₓ/∂x 和 ∂Fᵧ/∂y 分别描述对应分量沿各自方向的局部变化率。散度是单位面积的净向外通量密度。F=(Fₓ,Fᵧ) is a two-dimensional vector field. ∂Fₓ/∂x and ∂Fᵧ/∂y are local rates of the corresponding components along their coordinate directions. Divergence is net outward flux density per unit area.
留意这个区别Watch for this
散度是局部净通量密度，不能只看箭头长度。向量场散度与 KL 散度同名但定义不同。Divergence is local net flux density, not merely arrow length. Vector-field divergence and KL divergence share a name but have different definitions.
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