Page 176 - Intro to Tensor Calculus

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PART 2: INTRODUCTION TO CONTINUUM MECHANICS

In the following sections we develop some applications of tensor calculus in the areas of dynamics,
elasticity, fluids and electricity and magnetism. We begin by first developing generalized expressions for the
vector operations of gradient, divergence, and curl. Also generalized expressions for other vector operators
are considered in order that tensor equations can be converted to vector equations. We construct a table to
aid in the translating of generalized tensor equations to vector form and vice versa.
The basic equations of continuum mechanics are developed in the later sections. These equations are
developed in both Cartesian and generalized tensor form and then converted to vector form.

§2.1 TENSOR NOTATION FOR SCALAR AND VECTOR QUANTITIES

We consider the tensor representation of some vector expressions. Our goal is to develop the ability to
convert vector equations to tensor form as well as being able to represent tensor equations in vector form.
In this section the basic equations of continuum mechanics are represented using both a vector notation and
the indicial notation which focuses attention on the tensor components. In order to move back and forth
between these notations, the representation of vector quantities in tensor form is now considered.

Gradient

i
N
1
2
For Φ = Φ(x ,x ,... ,x ) a scalar function of the coordinates x ,i =1,... ,N , the gradient of Φ is
defined as the covariant vector
∂Φ
Φ ,i = , i =1,... ,N. (2.1.1)
∂x i
The contravariant form of the gradient is
g im Φ ,m. (2.1.2)
i
Note, if C = g im Φ ,m,i =1, 2, 3 are the tensor components of the gradient then in an orthogonal coordinate
system we will have
11
1
3
2
33
22
C = g Φ ,1 , C = g Φ ,2 , C = g Φ ,3 .
2
We note that in an orthogonal coordinate system that g ii =1/h ,(no sum on i), i =1, 2, 3 and hence
i
replacing the tensor components by their equivalent physical components there results the equations
C(1) 1 ∂Φ C(2) 1 ∂Φ C(3) 1 ∂Φ
= 2 , = 2 , = 2 .
h 1 h ∂x 1 h 2 h ∂x 2 h 3 h ∂x 3
3
2
1
Simplifying, we find the physical components of the gradient are
1 ∂Φ 1 ∂Φ 1 ∂Φ
C(1) = , C(2) = , C(3) = .
h 1 ∂x 1 h 2 ∂x 2 h 3 ∂x 3
2
These results are only valid when the coordinate system is orthogonal and g ij =0 for i 6= j and g ii = h ,
i
with i =1, 2, 3, and where i is not summed.