Vector Derivative

Question 1

scalar functions

Answer 1

像 w = f ( x,y,z )這樣, 賦予空間上每一點一個沒有方向的值

又稱為scalar field

Question 2

Arc length
弧線由很多個 r = < x,y,z > 組成
在scalar field上

Answer 2

一點點弧線的改變:
dr = < dx,dy,dz >
一點點弧線的改變長度:
|dr| = ( (dx)^2 + (dy)^2 + (dz)^2 )^(1/2) = ds (弧長)
弧線在r = < x,y,z >這點的切線 unit vector:
u = dr / | dr | 
   = < dx/ds,dy/ds,dz/ds >
|u| = ( (dx/ds)^2 + (dy/ds)^2 + (dz/ds)^2 )^(1/2) = 1

Question 3

direction derivative of f(x,y,z),

at point P, along direction u (it’s scalar function)

Answer 3

s是在funtion f上along direction u的一條線(弧線or直線)

df/ds
= (Df/Dx)(dx/ds) + (Df/Dy)(dy/ds) + (Df/Dz)(dz/ds)
=·
= ▽f · u (unit vector)

Question 4

direction derivative

Answer 4

df/ds —— funtion f對於所求 direction的derivative

Question 5

maximum direction derivative

Answer 5

df/ds
= ▽f · u
= |▽f| · |u| · cosθ
df/ds reach maximum when θ = 0, cosθ = 1, direction of gradient = direction of u
#u is the tangent unit vector of s at point P

Question 6

the direction that f(x,y,z) get the maximum change

Answer 6

= the direction that f(x,y,z) get the maximum direction derivative
= ▽f · u
= unit vector of gradient