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Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition
book

Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition

by Andrei D. Polyanin, Vladimir E. Nazaikinskii
December 2015
Intermediate to advanced content levelIntermediate to advanced
1643 pages
59h 33m
English
Chapman and Hall/CRC
Content preview from Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition
Chapter 1 2
Systems of Linear
Partial Differential Equations
12.1 Preliminary Remarks. Some Notation and Helpful
Relations
Let f and u = u e
1
+ v e
2
+ w e
3
be arbitrary sufficiently smooth scalar and vector func-
tions, and let e
1
, e
2
, and e
3
be the unit coordinate vectors corresponding to the Cartesian
coordinates x, y, and z. Then, by definition, w e have
f = f
x
e
1
+ f
y
e
2
+ f
z
e
3
,
div u = · u = u
x
+ v
y
+ w
z
,
curl u = ×u = (w
y
v
z
)e
1
+ (u
z
w
x
)e
2
+ (v
x
u
y
)e
3
,
f = f
xx
+ f
y y
+ f
zz
,
where the subscripts x, y, and z stand for the derivatives. The following differential rela-
tions hold:
curl f = 0, div curl u = 0, div f = f,
curl curl u = div u u, ∆(xf ) = x
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Publisher Resources

ISBN: 9781466581494