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Mathematical Techniques 2

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This Module is a continuation of Mathematical techniques 1, where we look at: complex analysis and integration; Fourier transforms; vector calculus; multi-dimensional integration; and integral theorems e.g. Green's theorem (Green was a miller and was the model mature student starting his studies wh...

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  • July 14, 2021
  • 69
  • 2020/2021
  • Class notes
  • Various
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MT2

, MT2 - Maths induction

Charlie, first half of lectures.

James, second half.


Pls do tutorial questions.

Each week new quiz for recap.

Schnums advanced calculus - books for this unit
Mary Boas general text book

Revision on vectors—
* vector colours .




A vector quantity has both direction 13 magnitude →
displacement
velocity .




A scalar only has magnitude / size → distance [will have a notation
speed a→ a- or a
]
Vectors can be written w.r.t, a basis which may density
,



temperature
be natural for coordinate system.

E.g cartesian coordinates- x,y,z,
= " " " " °
"




unit vector

magnitude = 1




magnitude of
a→ is →
/ I'M =
ai+aj
ñ

_g÷
given then
,


unit vector in the air
Ñ
is a. of


Eg ñ +3¥ foil
=
+
+5=145
.




=




IT =¥⇐÷rE+¥ñ

, Scalar product




B→
defined by
F. Ñ =
IF/ / BY -
Gso



⇐¢ =Ñ E -




it


A-Y.coso.fi?-5ii--r-i.--:i-.
this means if is a hit vector then


=/ AT / II. cosa =/
length of E. u→
projection onto
the air oefñ
if I p B→ perpendicular A?Ñ=o ( o will
Cesaro
are =
go ,

)

.IE/-qsg--la-?i(I+-B).E--A-?EtB?E
F- A- =/ AT
>




in cartisesn -




÷
-
i -
I
E. I j I E. I
- -




= .
= =\


F. I =
Jic = I = 0




Angle between vectors Cosa
Ñ¥;§-
=




Example
on
find angle between the
diagonal of the cube Pa side

=
i+J+É I;v÷ (F) IF I E) + + = 1+0.0--1

Ñ=
.




'

Ii? / =ti+=rs viii. ri l -
-




•so
ѵ,+¥- ¥ ¥ 0--65%1
-
-
=
=
=
,

, r




vector product
I'RE (IATIBT sinon

given vectors define another vector by XB = .




But still
÷;÷ : agnes
have I' x ⑤ E) +
.mmriatnE¥÷÷ = FIB
+ Axe
in cart'sear basis
I x K I XI '
-


I ex I




¥
a- x be =
I I k


÷ :÷ :: !
-

Az by AJ will be minus in the



:
on .




.am


I
=
flu by -


ay bx)

E g (I t
21) x (I t 5k ) I I i 2×5-3×0 101
tzj ±
-
.




I 2 O
j 5 - o 5J
-
i 3 5

K 3 2
I
-




tk
IOI -

5J
Tutorial sheet will have triple vectors -


F
/ ÷; )
-




scale rtripu product =
± LEXI =
Exit ) . . .
o
-
-




:L:÷÷÷÷i÷
Gc Any Azz
"


m. in
:* .

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