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View Full Document Waves: Mathematical preliminaries
The gradient operator:
(cartesian coordinates):
k
z
j
y
i
x
r
r
r
∂
∂
+
∂
∂
+
∂
∂
=
∇
When the gradient is applied to a
scalar
function U(x,y,z) the result is a
vector
called
the gradient of U:
k
z
U
j
y
U
i
x
U
U
r
r
r
∂
∂
+
∂
∂
+
∂
∂
=
∇
(1)
The components of the gradient on the axes of coordinates
)
cos(




θ
⋅
⋅
∇
=
⋅
∇
=
r
d
U
r
d
U
dU
r
r
Total differential of
U
:
(2)
Waves: Mathematical preliminaries
)
cos(




θ
⋅
⋅
∇
=
⋅
∇
=
r
d
U
r
d
U
dU
r
r
Some important properties of the gradient:
is directed along the steepest rate of increase of
U (cos(
)=1)
is perpendicular to the any surface
U(x,y,z) = const.
1.
2.
U
∇
U
∇
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View Full Document Waves: Mathematical preliminaries
The divergence of a vector function:
(cartesian coordinates):
k
z
y
x
E
j
z
y
x
E
i
z
y
x
E
z
y
x
E
z
y
x
r
r
r
r
)
,
,
(
)
,
,
(
)
,
,
(
)
,
,
(
+
+
=
z
E
y
E
x
E
E
z
y
x
∂
∂
+
∂
∂
+
∂
∂
=
∇
r
(3)
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This note was uploaded on 10/19/2009 for the course PHY 31 taught by Professor Cebe during the Fall '08 term at Tufts.
 Fall '08
 CEBE

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