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January 30th, 2017, 02:44 PM   #11
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Originally Posted by zylo View Post
z = f(x,y)

1) $\displaystyle dz = f_{x}dx + f_{y}dy = \bigtriangledown f\cdot$ dr

2) $\displaystyle dz/ds = \bigtriangledown f\cdot$ n

3) $\displaystyle f_{x}dx + f_{y}dy - dz = N\cdot$ dr = 0

1) Gradient. $\displaystyle \bigtriangledown f$ points in dir of max increase of z; see Eq 2).
2) Directional derivative in dir of unit vector n. n=dr/ds, ds=|dr|
3) Equation of tangent to surface. N is normal to surface.

dr is defined by respective Eq.
dz is not normal to the surface. See 1)
N is normal to the surface. See 3)
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