0 [sin(y)- 6) Let F(x, y) = x cos(y) - 2xy Is F conservative? If so, give a potential function for F. Given: So, F (21, y) = F(2₁ y) = (Sinly) - y² + ₁)² + (ucus (8) -zny) j Fi f₂ is to be Conservative for f(y) df₂ du Sin (y)- y² +1 ncos (4)-2uy Əf₂_a (ucos(y)-zny) = (1) Cosly) - 2y (1) au an = cos(y)-2y here, afz ди a fi ay a fi a (sinly) - y² +₁) _ = ay ay 7 = 24 afi ay = cos(y) -zy Hence, F is Conservative F (ny) = (Sinly)- y²+1)i + (n cos(y)-zuy) j for Potential function of f(414) S fata = S(+²23-8²+1) ² = (os(y) - 24 to = cos(y)-24 = Sinly) - y²n+n + c(y) (fy dy = {(u Cos (8) - 2xy) dy Sin(y) - 22-y² + c(u) = usin(y)-ny² + c(u) So, potential function flu,y) = (fadu + ffydy = 2 Sin(y)-ny² +2+ C {c=cly)+((a) plz [sin(y)- y² +11 7) Let F(x, y) = x cos(y)-2xy and let C be the curve (t², 2(n-t)), where 1st≤ 2. Use your work from question 6 to evaluate: F-dr
0 [sin(y)- 6) Let F(x, y) = x cos(y) - 2xy Is F conservative? If so, give a potential function for F. Given: So, F (21, y) = F(2₁ y) = (Sinly) - y² + ₁)² + (ucus (8) -zny) j Fi f₂ is to be Conservative for f(y) df₂ du Sin (y)- y² +1 ncos (4)-2uy Əf₂_a (ucos(y)-zny) = (1) Cosly) - 2y (1) au an = cos(y)-2y here, afz ди a fi ay a fi a (sinly) - y² +₁) _ = ay ay 7 = 24 afi ay = cos(y) -zy Hence, F is Conservative F (ny) = (Sinly)- y²+1)i + (n cos(y)-zuy) j for Potential function of f(414) S fata = S(+²23-8²+1) ² = (os(y) - 24 to = cos(y)-24 = Sinly) - y²n+n + c(y) (fy dy = {(u Cos (8) - 2xy) dy Sin(y) - 22-y² + c(u) = usin(y)-ny² + c(u) So, potential function flu,y) = (fadu + ffydy = 2 Sin(y)-ny² +2+ C {c=cly)+((a) plz [sin(y)- y² +11 7) Let F(x, y) = x cos(y)-2xy and let C be the curve (t², 2(n-t)), where 1st≤ 2. Use your work from question 6 to evaluate: F-dr
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Please help with question 6
Transcribed Image Text:6) Let F(x, y) = x cos(y) - 2xy
[sin(y)- y² +
Given:
50,
F (2₁8) =
df₂
an
F (2₁₁y) = (Sinly)-y²+₁) 2 + (ucus (8) -zuy) j
Fi
f₂
for f(y) is to be Conservative
a fi
2y
Əf₂
an
a fi
ay
Is F conservative? If so, give a potential function for F.
Sin (y)- y² +1
ncos (4) - 224
=
Ə (ncos (Y)-2ny) = (1) Cosly) - 2y (1)
=
du
= cos(y)-2y
2 (sinly) - y²+1) = cos(y) -24 +0
бу
= cos(y)-24
here, a.f₂
= af,
อน
ay
= cos(y) -zy
Hence, F is Conservative
F (219) = (Sin(y)- y²+1)i +
+ (n cos(y)-zuy) j
for Potential function of f(414)
Sfan = Sisontat-2²+Juk
- y² + 1) du
- y³²₁n +4 + c(y)
= Sinly) 4
ffy dy = f(u cos (8)-248) dy
= u Sinly) - 22.4² + c(u)
= usin(y) - uy² + c(u)
So, potential function
f(2₁9) = (fudu + [fydy
= ~ Sinly)-ny² +2 + c {c= cly)+cla)
7) Let F(x, y) = [sin(y)- y² +11
x cos(y)-2xy
Use your work from question 6 to evaluate:
plz
and let C be the curve (t², 2(n-t)), where 1 sts 2.
Sc
F.dr
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