PPT - Elements of Vector Calculus Electrical Engineering (EE) Notes | EduRev

Electromagnetic Theory

Electrical Engineering (EE) : PPT - Elements of Vector Calculus Electrical Engineering (EE) Notes | EduRev

 Page 1


In 1-D
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
In 3-D
Differential Calculus w.r.t. Space
Definitions of div, grad and curl
t 0
?S
1
div lim . dS
t
d
d
®
æö
º
ç÷
èø
ò
D Dn
Ñ
t 0
?S
1
curl lim dS
t
d
d
®
æö
º-´
ç÷
èø
ò
D Dn
Ñ
Elemental volume dt
with surface DS
n
dS
D=D(r), f= f(r)
[ ]
x0
1
lim ( ) ()
df
fx x fx
dxx
D®
æö
= +D-
ç÷
D
èø
Page 2


In 1-D
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
In 3-D
Differential Calculus w.r.t. Space
Definitions of div, grad and curl
t 0
?S
1
div lim . dS
t
d
d
®
æö
º
ç÷
èø
ò
D Dn
Ñ
t 0
?S
1
curl lim dS
t
d
d
®
æö
º-´
ç÷
èø
ò
D Dn
Ñ
Elemental volume dt
with surface DS
n
dS
D=D(r), f= f(r)
[ ]
x0
1
lim ( ) ()
df
fx x fx
dxx
D®
æö
= +D-
ç÷
D
èø
fndS
(large)
Gradient
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
dS
n
Elemental volume dt
with surface DS
fndS
(small)
fndS
(medium)
fndS
(medium)
= magnitude and direction of the slope in the scalar field at a point
Resulting fndS
Page 3


In 1-D
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
In 3-D
Differential Calculus w.r.t. Space
Definitions of div, grad and curl
t 0
?S
1
div lim . dS
t
d
d
®
æö
º
ç÷
èø
ò
D Dn
Ñ
t 0
?S
1
curl lim dS
t
d
d
®
æö
º-´
ç÷
èø
ò
D Dn
Ñ
Elemental volume dt
with surface DS
n
dS
D=D(r), f= f(r)
[ ]
x0
1
lim ( ) ()
df
fx x fx
dxx
D®
æö
= +D-
ç÷
D
èø
fndS
(large)
Gradient
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
dS
n
Elemental volume dt
with surface DS
fndS
(small)
fndS
(medium)
fndS
(medium)
= magnitude and direction of the slope in the scalar field at a point
Resulting fndS
Review
ò
®
º
?S
0 t
dS
t
1
Lim grad n f
d
f
d
Gradient
Magnitude and direction
of the slope in the scalar
field at a point
Page 4


In 1-D
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
In 3-D
Differential Calculus w.r.t. Space
Definitions of div, grad and curl
t 0
?S
1
div lim . dS
t
d
d
®
æö
º
ç÷
èø
ò
D Dn
Ñ
t 0
?S
1
curl lim dS
t
d
d
®
æö
º-´
ç÷
èø
ò
D Dn
Ñ
Elemental volume dt
with surface DS
n
dS
D=D(r), f= f(r)
[ ]
x0
1
lim ( ) ()
df
fx x fx
dxx
D®
æö
= +D-
ç÷
D
èø
fndS
(large)
Gradient
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
dS
n
Elemental volume dt
with surface DS
fndS
(small)
fndS
(medium)
fndS
(medium)
= magnitude and direction of the slope in the scalar field at a point
Resulting fndS
Review
ò
®
º
?S
0 t
dS
t
1
Lim grad n f
d
f
d
Gradient
Magnitude and direction
of the slope in the scalar
field at a point
Gradient
 Component of gradient is the partial derivative
in the direction of that component
 Fourier´s Law of Heat Conduction
s
s
¶
¶
= Ñ
f
f . e
n . T k
n
T
k q Ñ - =
¶
¶
- =
&
f Ñ
s
s e ,
Page 5


In 1-D
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
In 3-D
Differential Calculus w.r.t. Space
Definitions of div, grad and curl
t 0
?S
1
div lim . dS
t
d
d
®
æö
º
ç÷
èø
ò
D Dn
Ñ
t 0
?S
1
curl lim dS
t
d
d
®
æö
º-´
ç÷
èø
ò
D Dn
Ñ
Elemental volume dt
with surface DS
n
dS
D=D(r), f= f(r)
[ ]
x0
1
lim ( ) ()
df
fx x fx
dxx
D®
æö
= +D-
ç÷
D
èø
fndS
(large)
Gradient
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
dS
n
Elemental volume dt
with surface DS
fndS
(small)
fndS
(medium)
fndS
(medium)
= magnitude and direction of the slope in the scalar field at a point
Resulting fndS
Review
ò
®
º
?S
0 t
dS
t
1
Lim grad n f
d
f
d
Gradient
Magnitude and direction
of the slope in the scalar
field at a point
Gradient
 Component of gradient is the partial derivative
in the direction of that component
 Fourier´s Law of Heat Conduction
s
s
¶
¶
= Ñ
f
f . e
n . T k
n
T
k q Ñ - =
¶
¶
- =
&
f Ñ
s
s e ,
z y x
dxdydz
z
dxdydz
y
dxdydz
x ¶
¶
+
¶
¶
+
¶
¶
=
÷
÷
ø
ö
ç
ç
è
æ
¶
¶
+
¶
¶
+
¶
¶
=
º
®
®
ò
f f f f f f
d
f
d
f
d
d
k j i k j i
n
t
1
Lim
dS
t
1
Lim grad
0 t
?S
0 t
Face 2
Differential form of the Gradient
ò
®
º
?S
0 t
dS
t
1
Lim grad n f
d
f
d
Cartesian system
dy
dx
dz
j
i
k
P
Evaluate integral by expanding the variation in
f about a point P at the center of an elemental
Cartesian volume. Consider the two x faces:
f = f(x,y,z)
Face 1
dydz
dx
x
Face
) (
2
dS
1
i n -
÷
ø
ö
ç
è
æ
¶
¶
- »
ò
f
f f
dydz
dx
x
Face
) (
2
dS
2
i n +
÷
ø
ö
ç
è
æ
¶
¶
+ »
ò
f
f f
adding these gives
dxdydz
x ¶
¶ f
i
Proceeding in the same way for y and z
dxdydz
y ¶
¶ f
j dxdydz
z ¶
¶ f
k and we get , so
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