Home work 2 solutions are as follows:
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The confusion I have with this whoie continuum mechanics is the way we use operators especially the (multiplication sign). At a time its used as a scaler multiplication when am thinking of cross product at another time it is actually used as a cross product . It makes the whole thing difficult because am reading with the mind that am reading vecctors. I hope you can help me get out of this problem sir. I need clarification
OK, for simplicity, let us keep within the recommended text (Gurtin). If you go sufficiently wide you will get confused and that will not be your fault because each author sometimes takes the law into his or her hands and creates new rules for representation as he/she defines them.
The key to not get confused here is to know the character of the values you are dealing with: Between two vectors, a simple multiplication is the cross product. There is NO other meaning. When there is a circle surrounding the “multiplication” sign, you are dealing with a dyad creating tensor product. Between a tensor and a vector, there should be no symbol whatsoever. Some authors use a dot also to represent the operation of a tensor on a vector; but our main text is the more consistent rule. A dot between two vectors is the scalar product which creates a scalar out of the two vectors as you have been used to from your secondary school.
Okay sir,thank you sir
In your solution †o no 5 of homework 2, you used two different set of indicies for the components of the same vector. Any reason for that sir? Because if am ask †̥o solve that question time Ά̲̣̥πϑ over again ℓ̊ will not think of using different set of indices because its the same vector multiplying itself
For any vector, given the fact that every basis has a dual, there are always two ways to put it in component form: The covariant way or the contravariant way. For a second-order tensor, there are four dual bases. Consequently, there are four ways to express each second-order tensor in component form: Covariantly, contravariantly and two mixed tensor component forms. When you are solving a problem, let your convenience determine which one you choose. In the solution of problem 5, I simply decided to choose what was most convenient for me – causing the least amount of confusion in terms that come into the analysis. My way is by no means the only way to solve the problem. In fact, it is a good practice to use a different representation and see if you will still be able to arrive at the result.
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