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Mathematical Engineering
Tensor Algebra and Tensor Analysis for Engineers
With Applications to Continuum Mechanics
vonMikhail Itskov
1. Auflage
Springer 2012
Verlag C.H. Beck im Internet:www.beck.de
ISBN 978 3 642 30878 9
Zu Inhaltsverzeichnis
schnell und portofrei erhltlich bei beck-shop.de DIE FACHBUCHHANDLUNG
Chapter 2Vector and Tensor Analysis in Euclidean Space
2.1 Vector- and Tensor-Valued Functions, DifferentialCalculus
In the following we consider a vector-valued function x .t/ and a tensor-valuedfunction A .t/ of a real variable t . Henceforth, we assume that these functions arecontinuous such that
limt!t0
x .t/ x .t0/ D 0; limt!t0
A .t/ A .t0/ D 0 (2.1)
for all t0 within the definition domain. The functions x .t/ and A .t/ are calleddifferentiable if the following limits
dxdt
D lims!0
x .t C s/ x .t/s
;dAdt
D lims!0
A .t C s/ A .t/s
(2.2)
exist and are finite. They are referred to as the derivatives of the vector- and tensor-valued functions x .t/ and A .t/, respectively.
For differentiable vector- and tensor-valued functions the usual rules of differen-tiation hold.
1. Product of a scalar function with a vector- or tensor-valued function:
ddt
u .t/ x .t/ D dudt
x .t/ C u .t/ dxdt
; (2.3)
ddt
u .t/ A .t/ D dudt
A .t/ C u .t/ dAdt
: (2.4)2. Mapping of a vector-valued function by a tensor-valued function:
ddt
A .t/ x .t/ D dAdt
x .t/ C A .t/ dxdt
: (2.5)
M. Itskov, Tensor Algebra and Tensor Analysis for Engineers, Mathematical Engineering,DOI 10.1007/978-3-642-30879-6 2, Springer-Verlag Berlin Heidelberg 2013
35
36 2 Vector and Tensor Analysis in Euclidean Space
3. Scalar product of two vector- or tensor-valued functions:
ddt
x .t/ y .t/ D dxdt
y .t/ C x .t/ dydt
; (2.6)
ddt
A .t/ W B .t/ D dAdt
W B .t/ C A .t/ W dBdt
: (2.7)4. Tensor product of two vector-valued functions:
ddt
x .t/ y .t/ D dxdt
y .t/ C x .t/ dydt
: (2.8)
5. Composition of two tensor-valued functions:
ddt
A .t/ B .t/ D dAdt
B .t/ C A .t/ dBdt
: (2.9)
6. Chain rule:ddt
x u .t/ D dxdu
dudt
;ddt
A u .t/ D dAdu
dudt
: (2.10)7. Chain rule for functions of several arguments:
ddt
x u .t/ ,v .t/ D @x@u
dudt
C @x@v
dvdt
; (2.11)
ddt
A u .t/ ,v .t/ D @A@u
dudt
C @A@v
dvdt
; (2.12)
where @=@u denotes the partial derivative. It is defined for vector and tensorvalued functions in the standard manner by
@x .u,v/
@uD lim
s!0x .u C s,v/ x .u,v/
s; (2.13)
@A .u,v/@u
D lims!0
A .u C s,v/ A .u,v/s
: (2.14)
The above differentiation rules can be verified with the aid of elementary differentialcalculus. For example, for the derivative of the composition of two second-ordertensors (2.9) we proceed as follows. Let us define two tensor-valued functions by
O1 .s/ D A .t C s/ A .t/s
dAdt
; O2 .s/ D B .t C s/ B .t/s
dBdt
: (2.15)
Bearing the definition of the derivative (2.2) in mind we have
lims!0 O1 .s/ D 0; lims!0 O2 .s/ D 0:
2.2 Coordinates in Euclidean Space, Tangent Vectors 37
Then,
ddt
A .t/ B .t/ D lims!0
A .t C s/ B .t C s/ A .t/ B .t/s
D lims!0
1
s
A .t/ C s dA
dtC sO1 .s/
B .t/ C s dB
dtC sO2 .s/
A .t/ B .t/
D lims!0
dAdt
C O1 .s/
B .t/ C A .t/dB
dtC O2 .s/
C lims!0 s
dAdt
C O1 .s/
dBdt
C O2 .s/
D dAdt
B .t/ C A .t/ dBdt
:
2.2 Coordinates in Euclidean Space, Tangent Vectors
Definition 2.1. A coordinate system is a one to one correspondence betweenvectors in the n-dimensional Euclidean space En and a set of n real num-bers .x1; x2; : : : ; xn/. These numbers are called coordinates of the correspondingvectors.
Thus, we can write
xi D xi .r/ , r D r x1; x2; : : : ; xn ; (2.16)where r 2 En and xi 2 R .i D 1; 2; : : : ; n/. Henceforth, we assume that thefunctions xi D xi .r/ and r D r x1; x2; : : : ; xn are sufficiently differentiable.Example 2.1. Cylindrical coordinates in E3. The cylindrical coordinates (Fig. 2.1)are defined by
r D r .'; z; r/ D r cos 'e1 C r sin 'e2 C ze3 (2.17)
andr D
q.r e1/2 C .r e2/2; z D r e3;
' D8