53 Define mathematically the dot and cross product operations of two vectors

53 define mathematically the dot and cross product

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5.3 Define mathematically the dot and cross product operations of two vectors using tensor notation.
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5.7 Exercises 144 5.4 Repeat the previous exercise for the scalar triple product and vector triple product operations of three vectors. 5.5 Define mathematically, using tensor notation, the three scalar invariants of a rank-2 tensor: I , II and III . 5.6 Express the three scalar invariants I , II and III in terms of the other three invariants I 1 , I 2 and I 3 and vice versa. 5.7 Explain why the three invariants I 1 , I 2 and I 3 are scalars using in your argument the fact that the three main invariants I , II and III are traces? 5.8 Gather six terms from the Index about the scalar invariants of tensors. 5.9 Justify, giving a detailed explanation, the following statement: “If a rank-2 tensor is invertible in a particular coordinate system it is invertible in all other coordinate systems, and if it is singular in a particular coordinate system it is singular in all other coordinate systems”. Use in your explanation the fact that the determinant is invariant under admissible coordinate transformations. 5.10 What are the ten joint invariants between two rank-2 tensors? 5.11 Provide a concise mathematical definition of the nabla differential operator in Cartesian coordinate systems using tensor notation. 5.12 What is the rank and variance type of the gradient of a differentiable scalar field in general curvilinear coordinate systems? 5.13 State, in tensor notation, the mathematical expression for the gradient of a differen- tiable scalar field in a Cartesian system.
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5.7 Exercises 145 5.14 What is the gradient of the following scalar functions of position f, g and h where x 1 , x 2 and x 3 are the Cartesian coordinates and a, b and c are constants? f = 1 . 3 x 1 - 2 . 6 ex 2 + 19 . 8 x 3 g = ax 3 + be x 2 h = a ( x 1 ) 3 - sin x 3 + c ( x 3 ) 2 5.15 State, in tensor notation, the mathematical expression for the gradient of a differen- tiable vector field in a Cartesian system. 5.16 What is the gradient of the following vector where x 1 , x 2 and x 3 are the Cartesian coordinates? V = (2 x 1 - 1 . 2 x 2 , x 1 + x 3 , x 2 x 3 ) What is the rank of this gradient? 5.17 Explain, in detail, why the divergence of a vector is invariant. 5.18 What is the rank of the divergence of a rank- n ( n > 0 ) tensor and why? 5.19 State, using vector and tensor notations, the mathematical definition of the divergence operation of a vector in a Cartesian coordinate system. 5.20 Discuss in detail the following statement: “The divergence of a vector is a gradient operation followed by a contraction”. How this is related to the trace of a rank-2 tensor? 5.21 Write down the mathematical expression of the two forms of the divergence of a rank-2 tensor. 5.22 How many forms do we have for the divergence of a rank- n ( n > 0 ) tensor and why? Assume in your answer that the divergence operation can be conducted with respect to any one of the tensor indices.
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5.7 Exercises 146 5.23 Find the divergence of the following vectors U and V where x 1 , x 2 and x 3 are the Cartesian coordinates: U = ( 9 . 3 x 1 , 6 . 3 cos x 2 , 3 . 6 x 1 e - 1 . 2 x 3 ) V = ( x 2 sin x 1 , 5( x 2 ) 3 , 16 . 3 x 3 ) 5.24 State, in tensor notation, the mathematical expression for the curl of a vector and of
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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