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Calculus Chapter 4 R50,00   Add to cart
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  3. University Of The Witwatersrand (wits)
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  5. Mathematics II (MATH2014)

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Calculus Chapter 4
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This chapter includes notes on Line Integration, Path Integral in Vector Fields, Double and Repeated Integrals, Volume between Two Surfaces and Green's Theorem. Each section includes multiple worked examples. Notes were taken on an iPad

Last document update: 5 year ago

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  • August 18, 2019
  • August 20, 2019
  • 27
  • 2018/2019
  • Study guide

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CHAPTER 4
VECTOR
CALCULUS

INTEGRATION

,.
Consider same function 01 lxiy ) that exists in the Z plane
Where parametric equation
Ms
(x y) be
rlss
•

, can
expressed as a )
So we have 01 ( )
•




. \
x Z
y




lg
\ 1

.

New consider path C on the x -




y plane
which is parameterised by the variable It )

* We can new
integrate 01 an
the
path C

flx , y ) arz " " "




#I
*




Mt ) =

( xlt ) , ylt ) ) :
× (a)
←




,
.




yla )
f. #)
l.MY#lihikteEt:nmnnetYet
surface




x
t=b




Ib) ,
,
the




ylb )
beneath
area of
the
The
'
curtain
¢
'




b

fcads f HH ) dd÷
* =
0 at

a

, *
direction




*
a- b-



e.g. Integrate 01 =xiy2 from ) 11,0
to ( 0,1 )
along
the paths shown
'



.




*


for integration to work




(.cz
b- Co. 2) ° .
-

,
• C ,
the
it
,

is to
necessary express
, •
•
Cir parametric equations
'

, } as
c


y:
,




Q 2,0 )




Along as :
-




parametric farm
rlt ) =
a- + t( b- -


e) of a straight line




gip ;'
÷
"
pgaratmneterisf '




limits




:[6-
fcads
:




BHH )
fba ddttdt
=



b :c on )=a t.tt




ftp.tfgEY.jhmr#atuba
.




1
t
=
1




a :( 1,0 )
=
( l -

t ,t )




;':*
If
t O




III
Is
=




' "
antes
"
asayarattzn
=fz[t ¥+53 ]
1.
-




x= it



=f2[
2


( i tat
-



's ) -

( o )] Y2= A

rz ±
=
.

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