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Assume that F(x, y, z) = (e^y) i + x(e^y) j + (z + 1)(e^z)k is a vector field defined on a smooth curve C given by r(t) = ti + (t^2)j + (t^3)k for 0 ≤ t ≤ 1. (i) Using the fundamental theorem of calculus, evaluate the line integral ∫C F · dr. (ii) Show whether ∫C F · dr is independent of path.

Assume that F(x, y, z) = (e^y) i + x(e^y) j + (z + 1)(e^z)k is a vector field defined on a smooth curve C given by r(t) = ti + (t^2)j + (t^3)k for 0 ≤ t ≤ 1. (i) Using the fundamental theorem of calculus, evaluate the line integral ∫C F · dr. (ii) Show whether ∫C F · dr is independent of path.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Assume that F(x, y, z) = (e^y)
i + x(e^y)
j + (z + 1)(e^z)k is a vector field defined on a
smooth curve C given by r(t) = ti + (t^2)j + (t^3)k for 0 ≤ t ≤ 1.
(i) Using the fundamental theorem of calculus, evaluate the line integral
∫C F · dr.
(ii) Show whether ∫C F · dr is independent of path.

Assume that F(x, y, z) = e³i+ xe³j + (z + 1)e²k is a vector field defined on a smooth curve C given by r(t) = ti+t²j+t³k for 0 ≤ t ≤ 1.
Transcribed Image Text:Assume that F(x, y, z) = e³i+ xe³j + (z + 1)e²k is a vector field defined on a smooth curve C given by r(t) = ti+t²j+t³k for 0 ≤ t ≤ 1.
(i) Using the fundamental theorem of calculus, evaluate the line integral So F. dr. (ii) Show whether F · dr is independent of path.
Transcribed Image Text:(i) Using the fundamental theorem of calculus, evaluate the line integral So F. dr. (ii) Show whether F · dr is independent of path.
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