12.520 Geodynamics

Problem set 1

Problem 3 (10%)

For practice in Einstein summation, evaluate the numerical value of d_ijd_ijand d_ii , both for three-dimensional coordinate system (i,j =1-3) and a 4-D system (i,j=1-4) briefly explain what you are doing.

Solution:

The Kronecker delta function is defined as:

a) d_ijd_ij: For a 3D system:

For a 4D system:

Here, we have not written out the i¹j terms because they are all zero.

b) d_ii: For a 3D system:

For a 4D system:

Here, all the terms are included since the i¹j terms are not part of the sum.

As you may see, d_ijd_ijand d_ii , are the same. Summing over the index j of some tensor multiplied by d_ij is the same are replacing j by i (and then make the sum if necessary). SO, of course, d_ijd_ijbecomes d_ii !

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