2) a) Show that any vector field in the form of F(x, y, z) = f(x)i + g(y)j + h(z) k where f, g and h are continuous functions that depend only on one variable, is a gradient field. b) Evaluate the integral of the vector field F(x,y,z)= sinxi + In(y^2 + 1)j over the "crazy curve" parametrized by r(t) = t(t-1) et³+4 sint/(1+ cos² (4t - 7)) i+t(t²- (0,0) - 1) (tan t³ + 4 sin t)/(1+ e4t-7) j, t = [0,1] (0,1) and (1,0)
2) a) Show that any vector field in the form of F(x, y, z) = f(x)i + g(y)j + h(z) k where f, g and h are continuous functions that depend only on one variable, is a gradient field. b) Evaluate the integral of the vector field F(x,y,z)= sinxi + In(y^2 + 1)j over the "crazy curve" parametrized by r(t) = t(t-1) et³+4 sint/(1+ cos² (4t - 7)) i+t(t²- (0,0) - 1) (tan t³ + 4 sin t)/(1+ e4t-7) j, t = [0,1] (0,1) and (1,0)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
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Could you help me with number 2
![Ws18. Group_
1) Given F(x, y, z) = x³ i + 3yj - z4 k and the curve C
parametrized by r(t) = (sint, cost,t), t = [0, π] evaluate
F.idx and F. k dz
a)
b) F.T ds
c) Consider the closed curve C' obtained by the union of C and the
segment that joins the points (0,-1, π) and (0,1,0). Evaluate the
integral of F along C' and verify that it is equal to zero.
2) a) Show that any vector field in the form of F(x, y, z) = f(x)i +
g(y)j + h(z) k where f, g and h are continuous functions that
depend only on one variable, is a gradient field.
b) Evaluate the integral of the vector field F(x,y,z)= sinxi +
In(y^2 + 1)j over the "crazy curve" parametrized by
r(t) = t(t-1) et³ +4 sint/(1+ cos² (4t - 7)) i + t(t² —
1) (tan t³ + 4 sin t)/(1+ e4t-7) j, t = [0,1]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F304238c1-892d-4f66-90f4-8d5405bfbe58%2Ffaab6a28-3477-432d-8c66-30c839cbe24d%2Fgpaekz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Ws18. Group_
1) Given F(x, y, z) = x³ i + 3yj - z4 k and the curve C
parametrized by r(t) = (sint, cost,t), t = [0, π] evaluate
F.idx and F. k dz
a)
b) F.T ds
c) Consider the closed curve C' obtained by the union of C and the
segment that joins the points (0,-1, π) and (0,1,0). Evaluate the
integral of F along C' and verify that it is equal to zero.
2) a) Show that any vector field in the form of F(x, y, z) = f(x)i +
g(y)j + h(z) k where f, g and h are continuous functions that
depend only on one variable, is a gradient field.
b) Evaluate the integral of the vector field F(x,y,z)= sinxi +
In(y^2 + 1)j over the "crazy curve" parametrized by
r(t) = t(t-1) et³ +4 sint/(1+ cos² (4t - 7)) i + t(t² —
1) (tan t³ + 4 sin t)/(1+ e4t-7) j, t = [0,1]
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