Indicial Notation

32 Whereas the Engineering notation may be the simplest and most intuitive one, it often leads to long and repetitive equations. Alternatively, the tensor and the dyadic form will lead to shorter and more compact forms.

33 While working on general relativity, Einstein got tired of writing the summation symbol with its range of summation below and above (such as ^aijbi) and noted that most of the time the upper range (n) was equal to the dimension of space (3 for us, 4 for him), and that when the summation involved a product of two terms, the summation was over a repeated index (i in our example). Hence, he decided that there is no need to include the summation sign ^ if there was repeated indices (i), and thus any repeated index is a dummy index and is summed over the range 1 to 3. An index that is not repeated is called free index and assumed to take a value from 1 to 3.

34 Hence, this so called indicial notation is also referred to Einstein's notation.

35 The following rules define indicial notation:

1. If there is one letter index, that index goes from i to n (range of the tensor). For instance:

assuming that n = 3.

2. A repeated index will take on all the values of its range, and the resulting tensors summed. For instance:

3. Tensor's order:

• First order tensor (such as force) has only one free index:

other first order tensors ajbj, Fikk, £j,jkUjvk

• Second order tensor (such as stress or strain) will have two free indeces.

Dii

D22

Di3

D2i

D22

D23

D3i

D32

other examples Aijip, Sijukvk. • A fourth order tensor (such as Elastic constants) will have four free indeces.

4. Derivatives of tensor with respect to xi is written as ,i. For example:

~3x~ = ^ ~3x~ = Vi,i dxj = Vi,j "dick = Ti'j,k (1.45)

36 Usefulness of the indicial notation is in presenting systems of equations in compact form. For instance:

this simple compacted equation, when expanded would yield:

Similarly:

Ali = B11C11D11 + Bii C12D12 + B12C11D21 + B12C12D22

A12 = B11C11D11 + B11C12D12 + B12C11D21 + B12C12D22

A21 = B21C11D11 + B21C12D12 + B22C11D21 + B22C12D22

A22 = B21C21D11 + B21C22D12 + B22C21D21 + B22C22D22

37 Using indicial notation, we may rewrite the definition of the dot product and of the cross product

we note that in the second equation, there is one free index p thus there are three equations, there are two repeated (dummy) indices q and r, thus each equation has nine terms.

0 0

Post a comment