+ Surface integral: which measures flux
-- outward normal for surface S, e.g. a sphere --
-- closed surface integration (e.g. sphere); opposite: open surface (e.g. bowl)
Steps: Choose convenient coordinate system and dS
Find and equation for each surface (e.g. cube has 6)
Calculate for each surface (Note: , and can be in different coordinates!)
Substitute equation for each surface into (e.g. 6 integrals for a cube)
Example 1.13: Surface integration

Example 1.14: Gauss's law

Example 1.15: More on surface integration

+ Gradient of a scalar f (Sect 2.5): grad f or ( -- ``del'')
The steepest slope in 3D with direction, i.e. gradient is a vector

+ grad f or , -- max slope, its direction is perpendicular to constant f surface.
slope along =
rectangular coord.:

+ Divergence of , div or (del dot A) (Sect 2.6) is a scalar defined as the outward flux per unit volume
Divergence theorem (p. 48):
Divergence measures spreading of a vector field, e.g electric field from a point charge
everywhere is solenoidal
Example 1.16: Divergence theorem
Example 1.17: divergence theorem

+ Curl of or (del cross A) (Sect 2.8) is a vector defined as circulation per unit area; its direction is perpendicular to the area and obeys right hand rule
Stokes's theorem (p. 59):
Curl measures rotation of a vector field, e.g. magnetic field from a current
everywhere is irrotational or conservative
Example 1.18: Stokes's theorem
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HW #5	Due 2/8/07