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Applied Calculus of Variations for Engineers, 2nd Edition
book

Applied Calculus of Variations for Engineers, 2nd Edition

by Louis Komzsik
June 2014
Intermediate to advanced content levelIntermediate to advanced
233 pages
5h 42m
English
CRC Press
Content preview from Applied Calculus of Variations for Engineers, 2nd Edition
Differential geometry 121
and
v

2
11
u
2
+2Γ
2
12
u
v
2
22
v
2
=0.
This result is the decoupled system of equations of the geodesic, introduced
in Section 8.1, hence the vanishing of the geodesic curvature is indeed a char-
acteristic of a geodesic curve.
Finally, since the recent discussions were mainly on parametric forms, the
equation of the geodesic for an explicitly given surface
z = z(x, y(x))
is quoted from [8] for completeness’ sake without derivation:
(1+(
∂z
∂x
)
2
+(
∂z
∂y
)
2
)
d
2
y
dx
2
=
∂z
∂x
2
z
∂y
2
(
dy
dx
)
3
+
(2
∂z
∂x
2
z
∂x∂y
∂z
∂y
2
z
∂y
2
)(
dy
dx
)
2
+
(
∂z
∂x
2
z
∂x
2
2
∂z
∂y
2
z
∂x∂y
)
dy
dx
∂z
∂y
2
z
∂x
2
.
The formula is rather overwhelming andusefulonlyinconnectionwiththe
simplest surfaces. ...
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Publisher Resources

ISBN: 9781482253597