Vector field tools
Introduced| English | Meaning |
|---|---|
| divergence/daɪˈvɜːdʒəns/ | A vector operation that measures how much a field flows outward from each point. |
| curl/kɜːl/ | A vector operation that measures how much a field circulates around each point. |
| divergence theorem | divergence theorem |
A signal at −130 dBm
- A phone's GPS receiver works with signals near −130 dBm, about a tenth of a femtowatt. Designing for that starts with the shape of fields.
- Two operators describe that shape: how much a field flows out of a point, and how much it swirls. This lesson puts them into coordinates.
Divergence: outflow per volume
- The divergence 散度 $\nabla \cdot \mathbf{F} = \dfrac{\partial F_{x}}{\partial x} + \dfrac{\partial F_{y}}{\partial y} + \dfrac{\partial F_{z}}{\partial z}$ is the net outflow per unit volume at a point.
- Positive means a source, negative a sink. For an electric field it measures the charge density there.
F = (x², y², z²). What is ∇·F at (1, 2, 3)?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
$\nabla \cdot \mathbf{F} = 2x + 2y + 2z = 2 + 4 + 6 = 12$.
Curl: circulation per area
- The curl 旋度 $\nabla \times \mathbf{F}$ measures how much the field circulates around a point, per unit area, and about which axis.
- Static electric fields have zero curl. Magnetic fields around currents, and electric fields where the magnetic field changes, do not.
A static electric field has zero curl everywhere.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
With no changing magnetic field, $\nabla \times \mathbf{E} = 0$, which is why a potential $V$ exists with $\mathbf{E} = -\nabla V$.
The divergence theorem
- The divergence theorem 散度定理 says the total outflow through a closed surface equals the divergence added up over the volume inside:
- It lets you count sources by counting flux through the boundary, which is how Gauss's law works.
The divergence theorem equates
Difficulty 1/5 · 100 coins + a small bonus on first completion today
Outflow through the boundary equals the sources inside. The loop-and-flux version is Stokes's theorem, for curl.
Worked example: divergence from coordinates
- $\mathbf{F} = (3x^{2}, 0, 0)$: $\nabla \cdot \mathbf{F} = \dfrac{\partial (3x^{2})}{\partial x} = 6x$. At $(2, 0, 0)$ that is 12 per unit volume.
- At $(-2, 0, 0)$ the field still points along $+x$, since $3x^{2} \ge 0$, but the divergence is −12. The sign follows how the field changes, not which way it points.
F = (3x², 0, 0). At x = −2 the field points along +x. What is the divergence there?
Difficulty 1/5 · 100 coins + a small bonus on first completion today
$6x = -12$. The field is shrinking as $x$ increases there, so more flows in than out: a sink, even though the field points along $+x$.
Worked example: both routes on a cube
- Flux of that field out of the unit cube $[0, 1]^{3}$, by volume: $\displaystyle\int_{0}^{1} 6x\,dx = 3$ (the $y$ and $z$ integrals each give 1).
- By surface: only the face $x = 1$ carries flux, $3 \times 1 \times 1 = 3$. The face $x = 0$ has $F = 0$, and the other four lie along the field. Both give 3.
F = (3x², 0, 0). What is the total flux out of the unit cube [0,1]³?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
$\int_{0}^{1} 6x\,dx = 3$ by volume; by surface, only the $x = 1$ face carries flux, $3 \times 1 = 3$.
Symmetry chooses the surface
- A spherically symmetric source calls for a sphere, with area element $r^{2}\sin\theta\,d\theta\,d\phi$. A long straight source calls for a cylinder, with $r\,d\phi\,dl$.
- On the right surface the field is constant, so the flux integral becomes field × area. With no symmetry, there is no shortcut: integrate directly.
Match each source to the Gaussian surface its symmetry calls for.
Difficulty 2/5 · 140 coins + a small bonus on first completion today
The surface must keep the field constant (or zero) over each part. A cube's field has no such surface.
Three common slips
- Divergence from the field's direction. It depends on how the field changes, not where it points.
- Losing the $r^{2}$. The sphere's area element carries an $r^{2}$ that cancels a $\dfrac{1}{r^{2}}$ field.
- Gauss's law without symmetry. Without a surface of constant field, field × area is not valid.
On a sphere of radius r around a point charge, the flux is the same for every r.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
The field falls as $\dfrac{1}{r^{2}}$ while the area grows as $4\pi r^{2}$; the two cancel, leaving $\dfrac{q}{\varepsilon_{0}}$ for every sphere.
Recap
- Divergence: outflow per volume. Curl: circulation per area.
- Divergence theorem: flux out of a surface = divergence summed inside.
- Symmetry picks the surface; check by computing both routes.