Lecture notes Mathematics II (MATH2011A) - Calculus_Chapter_4
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Course
Mathematics II (MATH2011A)
Institution
University Of The Witwatersrand (wits)
This document clearly describes, with detailed notes and examples, how to evaluate/solve the following:
~ Path Integrals in Scalar Fields
~ Path Integrals in Vector Fields
~ Double and Repeated Integrals
~ Enclosed Volumes
~ Change of variables in Double Integrals
~ Green's Theorem - Corolla...
9- SCALAR FIELD IS 0
2 THE PATH TALK ABOUT IS LABELLED [
.
3 THE ARC LENGTH IS DENOTED BY S .
4
INTEGRAL OVER C OF ¢ W R -1
- .
.
5 !
f. ¢cx,y ds
ONE INTERPRETATION OF WHAT THIS REPRESENTS IS :
↳ AND IF IT IS A TWO
VARIABLE FUNCTION
AS WELL !
( 3D)
}
"" " """" """" " "" "
> C IS THE PATHWAY
>
¢ IS THE HEIGHT OF
" "
SO CALLED FENCE
% TO CALCULATE THIS
THE AREA OF
FENCE , WE WOULD CALCULATE THE
PATH INTEGRAL to
•
WE DO NOT WANT TO WORK WITH S -
ARC LENGTH AS THE VARIABLE
% WE PARAMETERISE TO CHANGE VARIABLES :
SUPPOSE TOOK dt :
£"
J }
ds dt REMEMBER IN "" N " " ER ^^ " C "
=
☐ ( r (f) dt
To to MUST REPRESENT THE INITIAL
POINT OF C AND t1 THE END
£0 POINT OF C.
PARAMETER /SEE
VECTOR n
fyot
DEPENDENT ON t T
" t T
µ
IF VARIABLE AS ARC LENGTH
dr
( r(É)
= 1 To USE + VE SIGN !
dt However .
tr ARCÉÉNGTHT
If 0 IF VARIABLE t As
!
•
N§ %USEVESlGNf
IT IS EASY TO SEE IF S 1ST I -
LOOK
AT THE DIRECTION OF THE CURVE /
THE DIRECTION OF INTEGRATION
!
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