37 A is a tensor of type m n and B is a tensor of type s t what is the type of

# 37 a is a tensor of type m n and b is a tensor of

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3.7 A is a tensor of type ( m, n ) and B is a tensor of type ( s, t ), what is the type of their direct product AB ? 3.8 Discuss the operations of dot and cross product of two vectors (see § Dot Product and Cross Product) from the perspective of the outer product operation of tensors. 3.9 Collect from the Index all the terms related to the tensor operations of addition and permutation and classify these terms according to each operation giving a short definition of each. 3.10 Are the following two statements correct (make corrections if necessary)? “The outer multiplication of tensors is commutative but not distributive over sum of tensors” and “The outer multiplication of two tensors may produce a scalar”. 3.11 What is contraction of tensor? How many free indices are consumed in a single contraction operation? 3.12 Is it possible that the contracted indices are of the same variance type? If so, what is the condition that should be satisfied for this to happen?
3.8 Exercises 94 3.13 A is a tensor of type ( m, n ) where m, n > 1 , what is its type after two contraction operations assuming a general coordinate system? 3.14 Does the contraction operation change the weight of a relative tensor? 3.15 Explain how the operation of multiplication of two matrices, as defined in linear algebra, involves a contraction operation. What is the rank of each matrix and what is the rank of the product? Is this consistent with the rule of reduction of rank by contraction? 3.16 Explain, in detail, the operation of inner product of two tensors and how it is related to the operations of contraction and outer product of tensors. 3.17 What is the rank and type of a tensor resulting from an inner product operation of a tensor of type ( m, n ) with a tensor of type ( s, t )? How many possibilities do we have for this inner product considering the different possibilities of the embedded contraction operation? 3.18 Give an example of a commutative inner product of two tensors and another example of a non-commutative inner product. 3.19 Is the inner product operation distributive over algebraic addition of tensors? 3.20 Give an example from matrix algebra of inner product of tensors explaining in detail how the two are related. 3.21 Discuss specialized types of inner product operations that involve more than one contraction operation focusing in particular on the operations A : B and A ·· B where A and B are two tensors of rank > 1 . 3.22 A double inner product operation is conducted on a tensor of type ( 1 , 1 ) with a tensor of type ( 1 , 2 ). How many possibilities do we have for this operation? What is the
3.8 Exercises 95 rank and type of the resulting tensor? Is it covariant, contravariant or mixed? 3.23 Gather from the Index all the terms that refer to notations used in the operations of inner and outer product of tensors.

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• Summer '20
• Rajendra Paramanik
• Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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