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Lecture notes Mathematics II (MATH2011A) - Calculus_Chapter_4 R350,00   Add to cart
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Lecture notes Mathematics II (MATH2011A) - Calculus_Chapter_4
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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...

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  • February 14, 2022
  • 41
  • 2021/2022
  • Class notes
  • Mensah folly-gbetoula/sameerah jamal
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CHAPTER 4 8 ECTOR INTEGRATION :


PATH INTEGRALS IN SCALAR FIELDS :


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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