Calculus Volume 3 · free preview
The Divergence Theorem
The divergence theorem converts the flux through a closed surface to the divergence's triple integral over the enclosed volume, the outflow's equality to the sources' total, the Green's flux form's space generalization.
1. The theorem
The theorem
2. Gauss's law and the inverse-square field
Gauss's law and the inverse-square field
3. The flux's computation and the incompressible
The flux's computation and the incompressible
4. The theorems' completion
The theorems' completion
5. Divergence theorem statement
The divergence theorem equates the flux through the closed surface S to which integral?
6. Radial field flux
The flux of F = (x, y, z) through the sphere of radius a equals which value?
7. Gauss law content
Gauss's law in electromagnetism is the divergence theorem's application to which field?
8. Incompressible flux
An incompressible field has zero divergence. The divergence theorem implies its flux through any closed surface is what?
9. Flux independence enclosing surface
The inverse-square field's flux through any sphere centered at the charge is the same. The divergence theorem implies this extends to which surfaces?
10. Flux outward convention
The divergence theorem uses the outward normal for the closed surface. As a direction, what is it? Answer with one word.
11. Interior faces
The proof's partition into small boxes has the interior faces' flux cancelling. As a two-word phrase, what faces? Answer like answer keys.
12. Gauss law constant
Gauss's law divides the enclosed charge by which constant? Answer with a single Greek-letter name.
13. Continuity equation
The fluid dynamics' continuity equation, the density's rate plus the divergence of the flux zero, expresses which conservation?
14. Inverse square divergence
Away from the origin, the inverse-square field's divergence is which number? Answer with a number.
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