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This document informs about electromagnetism
Typology: Lecture notes
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1
Gradient –
Divergence –
Curl
Divergence & Stokes’
Theorems
Null Identities –
Helmholtz Theorem
Assist. Prof. Dr.
Özlem Özgün Office^
:^ S- Phone^
:^ 661 2972 E-mail^
:^ ozozgun@metu.edu.tr Web^
:^ http://www.metu.edu.tr/~ozozgun/
2
T or A.
T or A.
(.)^
(.)^
(.) ˆ^
ˆ^
ˆ
x^
y^
z
a^
a^
a
x^
y^
z
∂^
∂^
∂
∇ =^
+^
∂^
∂^
∂
4
5
Question: Suppose we have a function of one variable:
What does the derivative,
df^ / dx
, do for us?
Answer: It tells us how rapidly the function
when we change the argument
In words: If we change
The derivative is theproportionality factor. The derivative
df^ / dx
is the
slope of the graph of
x
First think “ordinary derivative”.
7
I want to fly in such a direction that I will get warm as soon as possible. In what direction must I
fly?
Δx^ T = 15^ °C T = 25^ °C
Δx
T = 20^ °C
T = 22^ °C^ Δx
T = 12^ °C
BEE Δx T = 18^ °C
Δx
Directionofgradient
8
The gradient
direction of maximum
increase of the function
The magnitude |
slope (rate of increase) alongthis maximal
direction.
Directionofgradient
(Function, T is
the^ hill height)
Gradient of
(vector quantity
Æ^ has magnitude & direction)
x^
y^
z
10
From^ these maps,
we can infer the
slope of the Earth’s
surface,
and^ the direction of these slopes
as topographic
contours lie closer together where the surface is
very steep.
Steep cliff
11
h (x,y)^ Æ
A scalar field representing the
height
(elevation) of the Earth at some point
denoted by
coordinates x and y.
13 Gradient 1 ˆ^
ˆ^
ˆ
r^
z
V^
V^
V
V^
a^
a^
a
r^
r^
φ z φ ∂^
∂^
∂
∇^ =
+^
∂^
∂^
∂ ˆ^
ˆ^
ˆ x^
y^
z
V^
V^
V
V^
a^
a^
a
x^
y^
z
∂^
∂^
∂
∇^ =
+^
∂^
∂^
∂ 1
1 ˆ^
ˆ^
ˆ sin
V^ R
V^
V
V^
a^
a^
a
R^
R^
θ R
φ
∂^
∂^
∂
∇^ =^
+^
∂^
∂^
Cylindrical Coordinates:^ ( ∂ Cartesian Coordinates: ,^ , r^ z^ φ^ )^ Spherical Coordinates: ,^ , R^ θ^ φ^ (^ )
14
1
C C
BUT; If
C^1 C^2 REMEMBER
: In general;
Integral depends on path.
“gradient of a scalar field g(r)”
1
C C
B
C
C A^ r^
d
g^ r^
d^ g
r^
g^ r
Depends on onlythe end points
16
17
Variations in a vector field can be represented by FLUX LINES,which are directed curves indicating the direction of vector fields.
Electric field of a charge
Flux: the rate of flow across a surface
19
Consider a volume V enclosed by a surface S When the volume contains a“source”, there will be a net“outward flux” through S.^ (+ divergence)
When the volume contains a“sink”, there will be a net“inward flux
” through S. (-^ divergence)
20
Divergence (
)^ measures how much the vector A spreads out (diverges) from the point in question. Divergence measures the outflow per unit volume of avector field at a point. Divergence
is a local measure of outgoingness. Origin: source (+ divergence)
Origin: sink (-^ divergence)
