Vector calculus: fields, flux and the cheap way to count
Introduzido| English | Significado |
|---|---|
| gradient/ˈɡreɪdɪənt/ | the steepest-ascent vector of a scalar field |
| flux/flʌks/ | the through-surface integral of a vector field |
| divergence theorem | closed-surface flux equals the volume integral of the divergence |
| curl/kɜːl/ | the local spin of a vector field |
| divergence/daɪˈvɜːdʒəns/ | the net outflow source strength of a vector field, pointwise |
Heat, air and the theorems
- The amplifier's temperature field $T = x^2 + 2y^2 + 3z^2$: gradient 梯度 $\nabla T = (2x, 4y, 6z)$ - at (1,1,1) that is (2, 4, 6), magnitude 7.483 K/m; Fourier turns it into flux $q = -40\nabla T = (-80, -160, -240)$ W/m2.
48 both ways
- Flux 通量 of $F = (x^2, y^2, z^2)$ through the cube [0,2]^3: three far faces carry 16 each, three near faces nothing - total 48 by surface.
- Divergence theorem: $\int 2(x+y+z)\,dV = 2 \times 3 \times 1 \times 8 = 48$. Two methods, one number; six surface integrals replaced by one volume integral.
Which operator answers this?
Route each question to the operator that owns it.
T = x^2 + 2y^2 + 3z^2. The y-component of grad T at (1,1,1)?
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grad T = (2x, 4y, 6z) - the y component is 4y.
The curl of (-y, x, 0) is:
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Uniform spin about z at rate 2.
Match each theorem to its trade.
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Two theorems and two laws carry the unit.
Spin and circulation
- $G = (-y, x, 0)$: curl 旋度 $(0, 0, 2)$, circulation $2\pi r^2$ - $2\pi$ at r = 1, and Stokes says exactly that: $(0,0,2) \cdot (0,0,\pi)$.
- The field's own balance: $\nabla^2 T = 12$ - a uniform 480 W/m3 source at k = 40.
q = -k grad T with k = 40: the magnitude of q at (1,1,1) (one decimal)?
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40 x sqrt(4+16+36) = 40 x 7.483.
Flux of (x^2, y^2, z^2) through the cube [0,2]^3?
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16 per far face x 3; divergence theorem 2 x 3 x 1 x 8.
Circulation of (-y, x, 0) round the unit circle?
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2 pi r^2 = 2 pi at r = 1; curl (0,0,2) over area pi.
Three field slips
- Computing grad/div/curl by position instead of by definition.
- Using the divergence theorem on open surfaces. Patch, then subtract.
- Quoting flux without saying through what.
The divergence theorem applies directly to the flux through a grille with one panel missing.
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Open surface: patch it, integrate, subtract the patch.
Integrating -k Laplacian T over 0.0002 m3 gives the amplifier’s generated heat: 96 mW.
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480 W/m3 x 0.0002 m3 - compared against the datasheet’s 100 mW.
Recap
- Gradient (2,4,6) uphill; $q = -k\nabla T$ at 299 W/m2; $\nabla^2 T = 12$.
- Divergence 散度 counts sources: 48 = 48. Curl counts spin: $2\pi = 2\pi$.
- The theorems trade awkward integrals for easy ones - closed surfaces only.