Calculate the circulation, fa F. dr, in two ways, directly and using Stokes' Theorem. The vector field F = (7x – 7y + 2z)(i + j) and C is the triangle with vertices (0, 0, 0), (5, 0, 0), (5, 8, 0), traversed in that order. Calculating directly, we break C into three paths. For each, give a parameterization 7 (t) that traverses the path from start to end for 0
Calculate the circulation, fa F. dr, in two ways, directly and using Stokes' Theorem. The vector field F = (7x – 7y + 2z)(i + j) and C is the triangle with vertices (0, 0, 0), (5, 0, 0), (5, 8, 0), traversed in that order. Calculating directly, we break C into three paths. For each, give a parameterization 7 (t) that traverses the path from start to end for 0
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:Calculate the circulation, F. dr, in two ways, directly and using Stokes' Theorem. The vector field F
(7x – 7y + 22)(i +5) and C is the triangle
%3D
with vertices (0, 0,0), (5, 0, 0), (5, 8, 0), traversed in that order.
Calculating directly, we break C into three paths. For each, give a parameterization 7 (t) that traverses the path from start to end for 0 <t < 1.
On Ci from (0, 0, 0) to (5, 0, 0), 7(t) =
On C2 from (5, 0,0) to (5, 8,0), 7(t)
On C3 from (5, 8, 0) to (0, 0, 0), 7(t) =
So that, integrating, we have a F. dr =
So, F. dř =
Sa F. dr =
and so a F. dr =
Using Stokes' Theorem, we have
curl F =
So that the surface integral on S, the triangular region on the plane enclosed by the indicated triangle, is
Ss curl F. dà = s
H dy dx,
where a =
c =
and d =
Integrating, we get fc F · dr =
Ss curl F. dà =
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