Scalars, vectors and tensors

The acceleration vector is given as
The units that we will use in class are length L, time T, mass M and temperature °. The units of a parameter are denoted in brackets. Thus

Newton’s second law is a vectorial statement: where denotes the force vector and m denotes the mass (which is a scalar)

The dimensions of the force vector are the dimension of mass times the dimension acceleration
Pressure p, which is a scalar, has dimensions of force per unit area. The dimensions of pressure are thus

The acceleration of gravity g is a scalar with the dimensions of (of course) acceleration:

A scalar is a zero-order tensor. A vector is a first-order tensor. A matrix is a second order tensor. For example, consider the stress tensor τ.

The stress tensor has 9 components. What do they mean? Use the following mnemonic device: first face, second stress

The force per unit face area acting in the x direction on that face is the stress τyx (first face, second stress).

The forces per unit face area acting in the y and z directions on that face are the stresses τyy and τyz.

Here τyy is a normal stress (acts normal, or perpendicular to the face) and τyx and τyz are shear stresses (act parallel to the face)

Shear stresses similarly reverse sign on the opposite face face are the stresses τyy and τyz.

Thus a positive normal stress puts a body in tension, and a negative normal stress puts the body in compression. Shear stresses always put the body in shear.`

or as

or in index notation, simply as

where i is understood to be a dummy variable running from 1 to 3.

Thus xi, xj and xp all refer to the same vector (x1, x2 and x3) , as the index (subscript) always runs from 1 to 3.

LECTURE 1: SCALARS, VECTORS AND TENSORS

is a vector.

Dot or scalar product of two vectors results in a scalar:

In index notation, the dot product takes the form

Einstein summation convention: if the same index occurs twice, always sum over that index. So we abbreviate to

There is no free index in the above expressions. Instead the indices are paired (e.g. two i’s), implying summation. The result of the dot product is thus a scalar.

Two free indices (i, j) means the result is a second-order tensor

Now consider the expression

This is a first-order tensor, or vector because there is only one free index, i (the j’s are paired, implying summation).

That is, scalar times vector = vector.

Third-order Levi-Civita tensor.

cycle clockwise: 1,2,3, 2,3,1 or 3,1,2

cycle counterclockwise: 1,3,2, 3,2,2 or 2,1,3

otherwise

Vectorial cross product:

One free index, so the result must be a vector.

Then

Thus for example

= 1

= -1

= 0

a lot of other terms that all = 0

i.e. the same result as the other slide. The same results are also obtained for C2 and C3.

The nabla vector operator :

or in index notation

or in index notation

The single free index i (free in that it is not paired with another i) in the above expression means that grad(p) is a vector.
The divergence converts a vector into a scalar. For example, where is the velocity vector,

Note that there is no free index (two i’s or two k’s), so the result is a scalar.

or in index notation,

One free index i (the j’s and the k’s are paired) means that the result is a vector