The Differential Line Vector for Coordinate (微分行向量的坐标).pdf
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9/6/2005 The Differential Line Vector for Coordinate Systems.doc 1/3
The Differential
Displacement Vector for
Coordinate Systems
Let’s determine the differential displacement vectors for each
coordinate of the Cartesian, cylindrical and spherical coordinate
systems!
Cartesian
This is easy!
d r ⎡ d x d y d z ⎤
⎛ ⎞ ˆ ⎛ ⎞ ˆ ⎛ ⎞ ˆ
dx dx a + a + a dx
⎢⎜ ⎟ x ⎜ ⎟ y ⎜ ⎟ z ⎥
dx dx dx dx
⎣⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎦
ˆ
a dx
x
⎡⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎤
d r d x ˆ d y ˆ d z ˆ
dy dy ⎢ a + a + a ⎥dy
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
dy ⎢ dy x dy y dy z ⎥
⎣⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎦
aˆ dy
y
d r ⎡ d x d y d z ⎤
⎛ ⎞ ˆ ⎛ ⎞ ˆ ⎛ ⎞ ˆ
dz dz a + a + a dz
⎢⎜ ⎟ x ⎜ ⎟ y ⎜ ⎟ z ⎥
dz dz dz dz
⎣⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎦
ˆ
a dz
z
Jim Stiles The Univ. of Kansas Dept. of EECS
9/6/2005 The Differential Line Vector for Coordinate Systems.doc 2/3
Cylindrical
Likewise, recall from the last handout that:
ˆ
d ρ aρ d ρ
Maria, look! I’m starting to see a trend!
d r ˆ
dx dx a dx
dx
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