Vector field tools
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| divergence/daɪˈvɜːdʒəns/ | divergence |
| curl/kɜːl/ | curl |
| symmetry/ˈsɪmətri/ | symmetry |
| flux/flʌks/ | từ thông |
Two operators, one test
- A field has an outflow per volume (the divergence) and a circulation per area (the curl). EE02 met them; here they take coordinates.
- The symmetry test decides the tool: spherical → Gauss on a sphere; cylindrical → a coaxial cylinder; none → direct integration.
$F = (3x^{2}, 0, 0)$: $\nabla \cdot F = 6x$, so the unit cube carries volume $6 \times 0.5 = 3$ — and the surface route (only the $x=1$ face, flux 3) agrees ✓.
{exercise:uni-ee0711-cube}
For F = (3x², 0, 0), the flux out of the unit cube [0,1]³ is —
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
Volume route: ∫6x dx over [0,1] = 3. Surface route: only the x = 1 face carries flux (3·1·1); x = 0 gives 0; the other four are parallel to F. Both routes say 3.
A charged infinite cylinder calls for the Gaussian surface —
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
Cylindrical symmetry keeps E constant along the curved side of a coaxial cylinder, so the integral collapses to E·2πrl.
The two-route habit
- Every field answer in this course can be checked a second way: integrate the sources, or count the flux through a surface. One number, two routes, zero doubt.
{exercise:uni-ee0712-routes}
A source has no symmetry at all. The right tool is —
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
No symmetry means the surface integral does not collapse to E×A — the shortcut is gone and direct integration is the honest route.