Question

Multivariable Calculus
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Let C be an oriented curve in R³; f = f(x,y,z) a function and  F a vector field. Which of the following is true?
The Answer Key (without solution) is telling me the answer is D....

PROBLEM 10: Let C be an oriented curve in R3, f-f(x, y, z) a function and F a vector field. Which of the following are true (1) J F dr is independent of parametrization. (2) J F dr changes sign if the direction (i.e. orientation) of the curve is changed (3) Je fds changes sign if the direction (i.e. orientation) of the curve is changed (4) F dr is independent of path. (5) Je Vf dr is independent of path A. 1,3,5 B. 2,4,5 C. all D. 1,2,5 E. 1,2,4

I really beg you.. could you please explain the reasons behind why your answer(s) are true and others are false?
While exam is soon, I am really having hard time understanding the concept--fundamentals behind it.
I will promise to sincerely give you thank you comment and Thumbs Up

Answer 1

Sotution 1o) Given tbat het C be the oriented Cusve in R and F a vecto fieH F dr is independent of Paiometeizatrsn is tTue Because, e inte then this Statement is true coithout alfiatom But f we int eq rate ovev a vecko fie , then the tuoo parametviPation with neversed aientations Jeild the Same value differing by a sjn Thus the integated fath 's independent the inteqrated Pth s (2) Sf.dv changes di vecront.e,οι: en sig if the of the Cuve changod t s true

a. fn) d . 0 (3) J p ds Chanq ot the uve is chargod ^t is False Because, thic is a double integ rated n cohich both oxientations Changes and ntega Semains unchanged. F d s IndependOf path So, it s isa Bacause, this is true only F s Consex vative. vf .d7 is indepen den Path LLe unCt orn f=> F.. ▽f )fis Conseyvative

So The Correct Onsa s S,2 and 5 Option D is CoTrect