This preview shows pages 1–6. Sign up to view the full content.
EE468G NOTES (2)
Reading assignment: Chapter 1
Contents: Calculus of scalar and vector fields
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Spatial differentiation and integration of scalar and
vector functions
Outline:
Gradient, divergence, and curl operators
Volume, surface, and line integral
Theorems about integrals
Gradient:
Operated on a scalar function
Result is a vector
The magnitude of the gradient is the max rate of change
The direction is the direction of max rate of change
In Cartesian system: gradient of scalar function
(, ,)
fxyz
is
defined as
()
ˆˆˆ
grad ( , , )
, ,
xyz
fff
f xyz
a
a
a
∂∂∂
=∇
=
+
+
Contour of a scalar function
Gradient:
A vector
Magnitude is the rate of change
In the direction of max rate of
change.
Solution:
(1)
By definition
( )
( )
( )
()
yz
a
z
x
a
xy
a
z
yz
y
x
a
y
yz
y
x
a
x
yz
y
x
a
f
z
y
x
z
y
x
10
ˆ
5
3
ˆ
6
ˆ
5
3
ˆ
5
3
ˆ
5
3
ˆ
2
2
2
2
2
2
2
2
+
+
+
=
∂
+
∂
+
∂
+
∂
+
∂
+
∂
=
∇
(2)
The max rate of change at P(1,2,0) is given by
71
.
6
45
0
2
10
0
5
1
3
2
1
6
2
2
2
2
2
0
,
2
,
1
=
=
⋅
⋅
+
⋅
+
⋅
+
⋅
⋅
=
∇
f
(3) First we need to determine the direction of P1P2:
()(
)
22
ˆˆˆ
ˆ
ˆ
10
21
ˆˆ
ˆ
ˆ
ˆ
11
2
x
yz
x
y
lx
y
x
y
la
a
a
aa
al
l
=−
+
−
+
−
=
+
==
+
+
=
+
r
rr
The rate of change in the direction of P1P2 is:
[]
2
5
3
2
6
2
/
ˆ
ˆ
ˆ
10
ˆ
5
3
ˆ
6
ˆ
2
2
2
2
z
x
xy
a
a
a
yz
a
z
x
a
xy
a
F
y
x
z
y
x
l
+
+
=
+
⋅
+
+
+
=
⋅
∇
(0,1,2)
30 52
601
ˆ
10 2
l
fa
⋅+
⋅
⋅⋅
∇⋅
=
+
=
Example
: given
,,
3
5
f xyz
xy
y
z
=+
,
(1)
Calculate
f
∇
.
(2)
Calculate the max rate of change of
f
at point P(1,2,0).
(3)
Calculate the rate of change of
f
along the direction
from point P1(0,1,2) to P2(1,2,2) evaluated at P(1,2,0).
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document The gradient operation in cylindrical system:
ˆˆ
ˆ
z
fff
fa
a
a
z
ρφ
ρρ
φ
∂∂∂
∇=
+
+
∂∂
∂
The gradient operation in sphereical system:
ˆ
sin
r
ff
f
a
a
rr
r
θφ
θθ
∂
+
+
∂
Solution:
In Cartesian system:
y
z
y
x
a
z
f
a
y
f
a
x
f
a
f
ˆ
ˆ
ˆ
ˆ
=
∂
∂
+
∂
∂
+
∂
∂
=
∇
In cylindrical system:
()
,,
s
i
n
fz
ρ
=
sin
sin
ˆˆˆ
ˆ
sin
cos
a
a
a
φφ
+
=
+
In spherical system:
s
i
n c
o
s
fr
r
θ
=
sin cos
sin cos
sin cos
sin
ˆ
sin cos
cos cos
sin
r
r
rrr
a
a
r
aa
a
+
+
∂
=+
−
Example
: given
fxyz y
=
,
Calculate
f
∇
in Cartesian,
cylindrical, and spherical systems:
Divergence:
Operated on a vector function
The result is a scalar function.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 01/19/2012 for the course EE 461 taught by Professor Cambron during the Spring '11 term at Ohio State.
 Spring '11
 Cambron

Click to edit the document details