Summary in Applied Mathematics, Lecture notes of Applied Mathematics

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Chapter

Matrices

&

System Linear

tan

b

e

· B

=

(F

i.0];

Row S

↑

·

Transposi

CATs

aue;

a

an

b.

=

7b

13

=

14

A

=

(ab),n+

=

[are]

column s

row 1,

row

20lumn

201umn3 · Row Echelon

Form REF

· Matrix B 15

x

• 28/umn

·

Y

2

I

row

j

8

1

↑

[

is

I

[

8

·

Rank of matrix

diogonal

non-diogonal

[ : a,3,

rB

=

3

A

=

rCA

=

[

I

·

Identity

matric

·

Inverse matrices

.

[33]

I

1 8 ⑧

I

A

=

[is]

a

a-bc

[%

=3]

1

08/

·

symmetrical

matrix

aij

=

aji

I

18 ⑧

AX

=

B

boat

transpose

(

.

X

=

B

A

/inconsistent

go

solution

types

of sistem

consistent

uniquesolution

semua leadings

independent

xperlulet

·

singular

matrices

(noninvertible)

do not have Inverses

bat

ERF

many

solution

dependent

is perinlet

any

number

inconsistent

[!= 11]

ada free variable

(0)

no solution

·

Non-singular

(invertible)

have

leverses

buat ERO

[

65/5]

only

zero

have hilar

·

Gaussion Elimination

Method

=

leading

one

consistent and

Independant

[

!a|m]

unique solution

/

2

back

substitution

Lu

=

mn

One solution

·

Can

change

row

[!w(%]

many

consistentand

dependent

50/

all zero

Chapter

3:

Partial Derivative

·H(x,y)

=

fn(x,y)

=

only respect

t

·

-fdu+of.

d re

d

e

⑬

of

(n,y)

=

fy(n,y)

=

only

respect

82

to

y

8x

We bawah

M

y

dn

dy

=darab

d t

& E

B

Of

=

fr

t

t

6x

Mr.du+

e

&

f

an(ev]

=

e

copy

exponent

I

2

d

Iff

W

2 · Implicit

Ordinary

Differentiation

dan

(en'y

=

auen

separate

one

by

one

& solve

it

· Implicit

Function theorem

· Product

rule F(n,y,z)

= 0

f

=

u

v

Find

= F.

Fn,

Fy

of

=

ux

uY 0z

I

:Fy

8

x

u

y

F

2

·

Relative

maxima

& mini and

of

functions

of two

variables

D

=

fan

(43) fyy

(x,y)

[Fay(x,3)]

"

· Find

Fr. Fun, Fy.

Fay, Fay

·

Find

Foo

0, Fy=

·

Find D

=

fun

(x,y)

yy(h,y)

[Fay(a,y)]"

Chapter

to

Integral

·

Double

integrals

over

rectangular regions

Cf(a,ys

dydn and

mode

in radian

·

Double

integrals

over

Nonrectangular regions

Draw a

graph

Regonize

the view

region

from

a or

y

·

Double

integrals

in

polar

cO ordinates

SchuldA:

fcrcoscs,

sin(n)

rodo

n

=

rcOS &

y

=

VS1nQ

From

equ

make

it

equ

of

circle:

v

~

all

al

y

x

= r

2

view

from

view from

N

y

9xIS

n- aXIS

W

2

Got radius,

determine

~"I

a

the

a

value

3

Insert it

in formuld

3/

↑

Figuring

the

positive

&

negative

value

Chapter

5:

Applications

of

Double

Integral

· Area of the bounded

regions

· The Centroids

b

9c(n)

A

=()

dydn

1. I

dA

x =

in

Ind

A

N

99,(n)

view

from

x

• ax( 5 -

(ydA

·

sketch

graph:

region

Decide the view from

ly

axis

·

Surface

Area

·

Moments

&

Centers

of Mass

A(s)

=

(.(+a(n))

(ty(n,y))

IdA

Find

M, density

=

tCa

a

m

=

p(x,y)

da

Iy

=

Fy

2 Find

it

A(s)

=

C(d

fa+fy+

dA

x

=

M

=

I)apCn,y)

dA

sketch

graph

to determine

the

region

Find

t

y

=

mx

=

byp(x,y)d

S

f(a)

d

A

area

= 1

dyde

↑Conclude

volume

=

suatu

milal

center of mass

(n,j)

z

=

f(x,y)

is at

point

Chapter

Numerical

Methods

(mode In

radian)

·

Trapezoidal

Rule

b

h

1

tens

dx

=

[(+(n0)

2f(n))

..."28cnnc)

f(un))

U

·Determine

b, a

&

h

b

a

h

=

n > subintervals

2 Do the number line based on

the Intervals

3 substitute

in

the

formula

·

sempson's

rule

(more

accurate)

1

f(x)dx

=

h((f(w)

4f(a))

2f(n))

4f(1s)

..

2f(Mu -z)

4f(an -) +

17(nn)]

h

=

b

a

• 14242... 42411 (coefficients

·

Runge

Kutta method

Yn

• 1

=

yn

y(k

2k

2k,

ku)

h

=

nn

• 1

xn

(between

two

point]

k,

=

f(nn,Yn)

B

=

f(nnz

In,yn

1

hpr)

k,

=f(Xn

+1h,yn"

-hr)

ky

=

f(un

h,yn

hBz)