In Cartesian coordinates, a vector field takes the form F = 2rzi+2yzj + (x² + y²) k (b) Calculate the line integral of F along a straight-line path starting at the origin and ending at the point (a,b,c). This path has the parametric representation x = at, y=bt, z=ct (0 ≤t≤ 1). (c) Given that the point (a, b, c) could be anywhere, use your answer to part (b) to find the scalar potential function U(x, y, z) corresponding to F, such that F = -VU. (d) Hence, or otherwise, calculate the value of the line integral of F along a path defined by the parametric equations z = cost, y = sint, z = = t (0 ≤ t ≤ π). (Hint: You can use the potential function U calculated in part (c) to evaluate this line integral.)
In Cartesian coordinates, a vector field takes the form F = 2rzi+2yzj + (x² + y²) k (b) Calculate the line integral of F along a straight-line path starting at the origin and ending at the point (a,b,c). This path has the parametric representation x = at, y=bt, z=ct (0 ≤t≤ 1). (c) Given that the point (a, b, c) could be anywhere, use your answer to part (b) to find the scalar potential function U(x, y, z) corresponding to F, such that F = -VU. (d) Hence, or otherwise, calculate the value of the line integral of F along a path defined by the parametric equations z = cost, y = sint, z = = t (0 ≤ t ≤ π). (Hint: You can use the potential function U calculated in part (c) to evaluate this line integral.)
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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Transcribed Image Text:In Cartesian coordinates, a vector field takes the form
F = 2.rzi+ 2yz j+ (r² + y²) k
(b) Calculate the line integral of F along a straight-line path starting at the
origin and ending at the point (a, b, c). This path has the parametric
representation
I = at, y = bt, z = ct (0<t<1).
(c) Given that the point (a, b, c) could be anywhere, use your answer to
part (b) to find the scalar potential function U(r, y, z) corresponding
to F, such that F = -VU.
(d) Hence, or otherwise, calculate the value of the line integral of F along a
path defined by the parametric equations
I = cost, y = sint,
t (0 <t<7).
(Hint: You can use the potential function U calculated in part (c) to
evaluate this line integral.)
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