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analysis3.txt
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analysis3.txt
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# Irrotational
curl F = 0
---
# Solenoïdal
div F = 0
---
# Harmonic
Δφ = div (grad φ) = 0
---
# Source-free
∫∫(F, n)dσ = 0 for all closed surfaces S
---
# Hypothesis H
For any closed surface in D, the domain enclosed by S is completely in D. Exs: sphere, cube, inside of torus, half-space. Not-fulfilling are: R^3 minus some points, domain in between two concentric spheres.
---
# div (curl F) = 0
---
# curl (grad φ) = 0
---
# Second derivative test
f\_xx * f\_yy - f\_xy^2 > 0 && f\_xx > 0 => local min
f\_xx * f\_yy - f\_xy^2 > 0 && f\_xx < 0 => local max
f\_xx * f\_yy - f\_xy^2 < 0 => saddle point
---
# Boundary candidates
at a max && grad f(a) points outwards => max
at a max && grad f(a) points inwards => no extreme value
at a min && grad f(a) points outwards => no extreme value
at a min && grad f(a) points inwards => min
---