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