Quiz #1 30/30 congratulations

1) AL-AMER, AHMAD ADNAN MOHA

2) AL-AGEELI, AHMAD IBRAHIM

3) AL-GARNI, BANDAR HASSAN S

4) AL-ARJANI, ALI SAEED ABDU

5) AL-BUGMI, TURKI MAHDI SHA

6) AL-BARAK, MUHAMMAD ABDUL

7) AL-MARRI, ALI MUHAMMAD FA

8) AL-HASAN, KHALED MUHAMMAD

9) AL-MANSOUR, ABDUL-RHMAN M

10)MAKKI, EMAD AHMAD MUHAMMA

11)AL-MESHAL, SAMI MUHAMMAD

2 )an indefinite integral

MATLAB can find

1) a definite integral 1

0

2dxx

syms x

Int)x^2, x, 0, 1)

dxx2

syms x

Int)x^2, x)

Group # 1

Turki al bogmiTurad al hujileAhmad aquileEmad Makki

Groups

Problem 1

9.9 Line Integral Independent of the Path

Evaluate

.along the curve C

between )-3,-3) and )3,3)

C

dyydxx 22

)-3,-3)

)3,3)

Problem 2

9.9 Line Integral Independent of the Path

Evaluate

.along the curve C between

)-3,-3) and )3,3)

C

dyydxx 22

)-3,-3)

)3,3)

Problem 3

9.9 Line Integral Independent of the Path

Evaluate

.along the curve C

between )-3,-3) and )3,3)

C

dyydxx 22

)-3,-3)

)3,3)

The Integral has the same value C

xdyydx

The integral

Is independent of the path

C

xdyydx

)-3,-3)

)3,3)

)-3,-3)

)3,3)

)-3,-3)

)3,3)

Under what condition the integral

is independent of the path C

QdyPdx

C

QdyPdxis independent of the path

QdyPdx is an exact differential

Test for exact differential

QdyPdx is an exact differential

x

Q

y

P

Problem 4

Which line integral is dependent of the path

C

dyxxydxxyy )32()66( 22

C

dyyxdxyx )5()2( 32

C

dyyxdxxy )42()32( 22

A)

C)

B)

Example3

Example4

HW 7

Problem 5

Application (1)

Evaluate

.along the curve C between

)-3,-3) and )4,4)

C

xdyydx

)4,4)

)-3,-3)

)4,0)

)0,-3)2)4(

16

3 xy

Problem 6

Evaluate C

xdyydx

Application (2)

)4,4)

)-3,-3)

)4,0)

)0,-3)2)4(

16

3 xy

.along the curve C between

)-3,-3) and )4,4)

Exact differential

QdyPdx is an exact differential

There exists a function ),( yx Such that

QdyPdxdyy

dxx

d

xdyydx xyyx ),(xdyydxd

Example

Px

Q

y

Theorem 9.8 Fundamental Theorem for Line Integral

Suppose there exists a function

such that ;that is,

is an exact differential. Then

depends on only the endpoints A and B

of the path C and

),( yx

QdyPdxd QdyPdx

C

QdyPdx

)()( ABQdyPdxC

Application (1)

Evaluate

.along the curve C between

)-3,-3) and )4,4)

C

xdyydx

)4,4)

)-3,-3)

)4,0)

)0,-3)2)4(

16

3 xy

xy

How to find ),( yx

QdyPdx dyy

dxx

)(y wrt e1)Integrat xgQdyφ

Qy

Qxg

)('x

wrt x derivative 2)Take

),( find then )( 3)Find yxxg

Method 2

)( wrt x e1)Integrat ygPdxφ

Px

Qyg

)('y

y wrt derivative 2)Take

),( find then )( 3)Find yxyg

Method 1

Which method ????? Easy step 1

C

dyxxydxxyy

yx

)32()66(

),( Find :4 Example22

C

dyxxydxxyy )32()66(

Evaluate :4 Example22

xyxxy 63

:found We22 )-1,0)

)3,4)

C

Notation

IfIs independent of the path between the endpoints A and B, then we write

C

QdyPdx

B

AQdyPdx

)4,4(

)3,3( :Example xdyydx

Theorem 9.10 Test for Path Independence

RdzQdyPdxC

Is independent of the path

x

Q

y

P

x

R

z

P

y

R

z

Q

How to find ),,( zyx

RdzQdyPdx

dzz

dyy

dxx

Evaluate. (2,1,4). and

(1,1,1)between Cpath any oft independen is

)19()3()(

that how :6 Example22

C

dzxyyzdyxzzxdxyzy

S

Conservative Vector Fields

QjPiF :force jdyidxdr )()( :disp

QdyPdxdrFW :WORK

If the Is independent of the path, then

2) F is said to be a gradient field

3) F is said to be conservative

4) Is a potential function for F

C

QdyPdx

F 1)

Conservative Vector Fields

In a gradient force field F,

1) The work done by the force upon a particle moving from position A to position B is the same for all paths.

2) The work done by a force along a closed path is zero

In a conservative field F,

1) The law of conservation of mechanical energy holds.

2) For a particale moving along a path in a conservative field,

kinetic energy + potential energy = constant

)0( 0 curl veconservati is FFRkQjPiF

WHY????

RdzQdyPdxC

Is independent of the path

x

Q

y

P

x

R

z

P

y

R

z

Q

ky

P

x

Qj

x

R

z

Pi

z

Q

y

RcurlF )()()(

RQPzyx

kji

FFcurl

9.9

9.7

Conservative Vector Fields

Conservative Vector Fields

Remarks (pp501)

A frictional force such as air resistance is neoconservative. Neoconservative forces are dissipative in that their action reduces kinetic energy without a corresponding increase in potential energy. In other words, if the work done depends on the path , then F is neoconservative.

9.9 Homework

)6,3(

)2,1(

22 )42()3(2y )7

points ebetween thpath convenient

any along integrate (b) and ,

such that function a find (a) : waysin two Evaluate

path. theoft independen is integralgiven that theshow

dyyxdxx

QdyPdxd

9.9 Homework

jyxiyxyxF

F

)()(),( )15

.for function

potential a find so, If field.gradient a is

fieldor given vect he whether tDetermine

33

9.9 Homework

)3,ln2,2(

)3,ln1,1(

222 23 )23

Evaluate path. theof

t independen is integralgiven that theShow

dzxedyydxe zz

9.9 Homework

curve. indicated thealong

)4()2(),(

force by the done work theFind 18)

jxeyiexyxF yy

)2,0))-2,0)

194

22

yx

9.9 Homework

ve]conservati is that show :[Hint /2).(0,2,

to(2,0,0) From ).3/,3(1, to(2,0,0) from

)()sin2()cos2()(helix thealong acting

)4()12()8(),,(

force by the done work theFind 28)32223

F

kzjtittr

kyxjzyxizxyzyxF