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รรรร =?
Resultant Vector
tan
And 12 ห ( 6 )ห
Let 6 ห 7 ห
2 2
x
y
A B
B x y
A x y
A
B
A + B
ฮธฮธฮธฮธ
System of Equations
()*+,-./
!0*1 2-
4 5
6 7$ $ 8
( 6 )( 3 ) ( 7 )( 4 ) 18 28 46 4 3
6 7 = โ โ =โ โ =โ โ
Determinant=
Larger Systems
det
15 12 10
60 15 2
12 7 10
3434 5 12
9 15 10 5 10
9 2 7 12 10
15 2 6 5 12 10
9 15 2
6 7 10
det
โ
=โ
โ
โ
โ โ โ
= โ
=โ
x
/*+.<1.>-./ + โ +
โ + โ
Spherical Coordinates
Cartesian coordinates: x, y, z Spherical coordinates: r, ฮธ , ฯ
( )
= + + = + +
=
=
=
โ
โ
r
z
x
y
r x y z x y z
z r
y r
x r
1
1
2 2 2 2
1 2 2 2
cos
tan
cos
sin sin
cos sin
ฮธ
ฯ
ฮธ
ฯ ฮธ
ฯ ฮธ
Cylindrical Coordinates
Cartesian coordinates: x, y, z Spherical coordinates: r, ฯ , z
( )
z z
x
y
r x y x y
z z
y r
x r
=
= + = +
=
=
=
โ 1
2 2 2
1 2 2
tan
sin
cos
ฯ
ฯ
ฯ
Partial Derivatives
So what is the
difference between โd โ and?
โ d โ like d/dx means the
function only contains the variable x.
When the function
contains not just x but may be y and z , we use the partial differential,
โ
โ
xyz yz x x
f
f x y z xyz
Forexample
= โ
โ
โ
โ
=
( )
( , , )
:
Note that the variables y and
z are held constant when the
differential operator acts on
the function
What is the solution to?
( )? ( )=? โ
โ
โ
โ xyz z
xyz y
โCrossโ or Vector Product
sin ฮธ
( )ห ( )ห ( )ห
ห ห ห
A B A B
A B ab ab x ab ab y ab ab z
b b b
a a a
x y z
A B
y z z y xz z x x y y x
x y z
x y z
ร =
ร = โ โ โ + โ
ร =
ฮธฮธฮธฮธ is the angle between A and B
First application of โdelโ: Gradient
โ f =โ( xyz )= yzx ห+ xzy ห+ xy z ห
z z
U y y
U x x
U F U ห ห ห
โ
โ + โ
โ
โ + โ
โ
โ = โโ = โ
The Scalar Product and โ
We can apply โ to the scalar product i.e.
โยท A where A is some vector
โยท A is called the โdivergenceโ of A or โdiv( A )โ.
Geometrically, we are discussing if A is diverging from some central point.
A is diverging from a central point so Div( A ) is equal to some value
A is not diverging from a central point so Div( A ) is equal to zero
The Vector Product and โ
We can apply โ to the vector product i.e.
โx A where A is some vector
โx A is called the โcurlโ of A or โcurl( A )โ.
Geometrically, we are discussing if A is curling around some central point.
A is not curling around a central point so curl ( A ) is equal to zero.
A is curling around a central point so curl ( A ) is equal to some value
Integrating over a closed loop
The loop can be circular or rectangular.
B โ
ds = B (^2 ฯ r^ )
From 0 to 2 ฯ
A B
D C
โLoopingโ from Point A to Point D
using straight line segments
d s = rd ฮธ ฮธ ห
ds = dsA x ห^ + dsBy ห+ dsCx ห+ dsD y ห
Closed Surface Integral
ฮธ ฮธ ฯ
ฮธ ฯ ฯ
R d d
da R d Rd
sin
( sin )( )
=^2
=
( )
2 E โ
da = E^4 ฯ^ R
The vector n-hat is
normal to the surface.
This means that โ da โ
must consist of the
differential distance in
the phi direction
multiplied by the
differential distance in
Theta is integrated from 0 to ฯ the theta direction, so and phi is integrated from 0 to 2 ฯ
Therefore if E depends only on R ,
then
da da n
First
E da
= ห
โ
