Handwritten Lecture notes of Engineering Mathematics

me

Engineering Mathematics me




2C



SEQUENCES & SERIES MATRICES & VECTORS f ,
cos
-
'




(Yy )
TO
Algebraic equations Ax =D
Linear
•



)
A¥r^
:

T
Sn
Effort ( 1) d) Linear Ordinary Differential Equations
=
•
:

Sn = n -




Acct)
Geometric series
day =




Arithmetic series a
•

Non Linear -
ODE 's : ITF =
ACK ) >act)
in Common Ratio
common
Difference Linear
Algebra
•




•

Example 1. Divide
A matrix
symmetric if it is equal
is - to its transpose .




2k2t3k-1_ by highest Skew
symmetric matrix if equal
its to its
transpose
• (k ) =
power g-
he negative .




7-k 't -4kt 2
Look
Every
2 at how he → N

square matrix A car be written
.


a Btc
2€72
-
-




x (k ) =
⇒ lim
-

-

If ⇒
converges as the sum
of asymmetric
B -
-

ELATA) Qlx ) -
x'Ax

7- +
I + I 9- C- ECA A
'


)
limg
-




B
=
matrix and skew matrix C
w
g
-
r
symmetric .




Infinite Arithmetic series win
always be
divergent Quadratics Cross
.




product

Any quadratic function be
•
can
- represented
for Convergence /Divergence
•
Tests
Q (x, .kz/=d..x2tdzzx2td.zxpcz as the sum
9-
:




Comparison test A series terms is
•




(dye d)(⇒
of positive Qcx ,xz ) Gc )
:

convergent xz d. ,
dig flat x'Act b'xtc
- -
-
-

, ,


if the value of
each
of its terms is less than
equal &



Lifdggvm
or "
↳
constant
the
corresponding terms
of another series of positive qufaordmatic
terms is
convergent .




①(" '' K2, " 3) =D" 392 + d22%22+133×32 +
dizcpcztdizk xztdzz >
czx,
A series
,
terms is
of positive divergent
if the value of
each
of its
greater than
terms is -




÷
co
equal "" "" " " "" Example
or

" " " d. 3
A- Btc

in:
-




snagging
eem.a-momeiesorros.me

Lima (AIT) CI
-


convergence
.
ai;D in?
Ratio
-




°
test : positive terms


hirsuta; sirirogona Symmetric :B ICAN 't { {I's ⇒ feed} f g) =L's:L)
-
= -


+ -

- ,

/AYLOR SERIES
Linear dependence
-

:

"


Cnbc ) wi
'

wi flew
flu) 1. Set Co f Ca)
is
independent
= a
of



(
-




gives
-




WII ! 5263¥
sea ,




7)
-


o , z z
wi -2W, 't wi
2-
Differentiate
⇒ Wsis linearly
'




gives a
Czfx apt
wrt

Cot C. ( a) ,
x

flat
=
-
- x -
+ - . . .
y s s

Wi & wz
'


( 47 ) dependent on
Iz f Wj
4 7 13
(z (a) B
"
- 10 = to
"

4=24f- G)
-




(

d÷⇒=
'
x a)
C. +24 (x a) +3 ↳ ( Maximum
t

=tff' linearly independent
number vectors in matrix
-
''

( )
-

a
- - -




↳ of
-


a


the number in its
is
equal to
of
non zero rows row echelon form
'T Fla)
- -
.




Cri
⇒ C, f' (a) ⇒
fact Ii Fca )Gc at
-

"

fi) 'm .&aij8i
- -




Determinant 8
- -


.
=
-




; ;
-




,


Inverse Matrices Pseudo Inverse non
square
Example of
a
-
• -




MACLAURIN SERIES fcxj-s.mx
•



matrix

G) adj
'

A-
fhcy.sn?Eont..fnCohcn
I Write out derivatives
= CA ) Given matrix A. matrix B
general Mxn
specify a
.
.




( )
"
sink
fix ) f
x =

that makes
-


sin >c
-




AB Im
:
-




BA In
- -
-
or
-

""
'

(" ' f COS"
Starling point (number of rows lessthan
=



f- cos"
-
'

is
always zero • When men number
of columns ) ⇒
right
is
2 put pseudo inverse of size nxm
defined as :
Art A'(AA )
-




equation
i

AArt=Im
-
.



into ⇒
=
. x=a

"

Maclaurin ) and k
for fck Taylor about When (number grows than columns) ⇒
-

x •
-

m >n
greater left pseudo

Taylor Maclaurin '
inverse size
of nxm is
defined
- -


as
Act (A'A)
'
= A ⇒
AEA In
Maclaurin :
f-( 2)
"
flat O '
I
-
=


f' x'
'
-




Example
'


fix , fcoytf Cola
361A
= t


f'(a)
+ - --


2nA -2×3 FAA )
"


f (a)
A
Right Art
'


⇒ A'
'
o
,
pseudo
2g
man ⇒
-
=

't
=

't
=




Is, ¥ , :X
sink -
- Ot k to -
. . - t


f (a)
''

la )
'' ,
O = ,
f
- -
-




1. Calculate transverse
f- Cal
'''


f. (a)
a.at?I .int
' '' '
= -

I
-
-
O 2- Calculate Inverse
fan ,
.

.




II
"
D-
"" '


Taylor caner .

ajaacaa 's
⇐ ?" "
"

's
feet sine 1 +
3. Do Pseudo Inverse
-
= -




¥1
. -
.




-3¥
6 !

det "" II
a
" = ,




" ⇒ -



E in -
'
n a'can 't
'




÷:
-
.