Vector Fields
Time Derivative

Derivatives of vectors are by
component.
d
ai  bi   dai  dbi
dt
dt
dt

Derivatives of vector
products use the chain rule.
d
sai   ds ai  s dai
dt
dt
dt
• Scalar multiplication
• Inner product
• Vector product
db j
dai
d
aibi   bi  a j
dt
dt
dt

d
 ijk aib j ek 
dt
db j 
dai 
  ijk
b j ek   ijk ai
ek
dt
dt
Normal Vectors
d
vi vi   0
dt
d
vi vi   dvi vi  vi dvi
dt
dt
dt
d
vi vi   vi ai  vi ai  2vi ai
dt
Example
 Show that the velocity of a
particle at constant speed is
normal to acceleration.

Use the inner product.
• Defines magnitude
• Commutes

Since viai = 0, vector v is
normal to a.
Space Derivative

Spatial derivatives depend
on the coordinates.
y

The partial derivatives point
along coordinate lines.
y = const.
The del operator is not a
vector but acts like one.
• Gradient changes scalar to
vector

x
x
• Not the same as the
coordinates.


y
x = const.
 
  ei
xi
i 

xi
 s
s  ei
xi
Vector Field

A vector field is a vector that
depends on position.

The differential operator is a
vector field.
• Acts on a scalar field
• Measures change
 
a  ar , t 
  
r   ei
xi
From Wolfram’s Mathworld
Divergence

The inner product of the del
operator with a vector is the
divergence.
• Scalar result

The divergence of a gradient
is the Laplacian.
• Del squared operator

Divergence is related to the
flow from a volume.

a
  a   i ai  i
xi

  s  
xi
 s 

   2 s
 xi 
2
2
2
  2

2
2
x1 x2
x3
2
 n v dS    a dV
i i
R
i i
R
Curl


  a   ijk  i a j ek

    a    k  ijk  i a j 

• Vector result
  ijk  k  i a j   kji  k  i a j
 ijk  k  i a j   kji  i  k a j

    a   0
but
 t a ds   
i i
S
S
n  j ak dA
ijk i
The vector product of the del
operator with a vector is the
curl.

The divergence of a curl is
zero.

Curl is related to the inner
product with the tangent
vector t.
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