24 State in tensor notation the mathematical expression for the curl of a

24 state in tensor notation the mathematical

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5.24 State, in tensor notation, the mathematical expression for the curl of a vector and of a rank-2 tensor assuming a Cartesian coordinate system. 5.25 By using the Index, make a list of terms and notations related to matrix algebra which are used in this chapter. 5.26 Define, in tensor notation, the Laplacian operator acting on a differentiable scalar field in a Cartesian coordinate system. 5.27 Is the Laplacian a scalar or a vector operator? 5.28 What is the meaning of the Laplacian operator acting on a differentiable vector field? 5.29 What is the rank of a rank- n tensor acted upon by the Laplacian operator? 5.30 Define mathematically the following operators assuming a Cartesian coordinate sys- tem: A · ∇ A × ∇ (239) What is the rank of each one of these operators? 5.31 Make a general statement about how differentiation of tensors affects their rank dis- cussing in detail from this perspective the gradient and divergence operations. 5.32 State the mathematical expressions for the following operators and operations assum- ing a cylindrical coordinate system: nabla operator, Laplacian operator, gradient of a scalar, divergence of a vector, and curl of a vector.
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5.7 Exercises 147 5.33 Explain how the expressions for the operators and operations in the previous exercise can be obtained for the plane polar coordinate system from the expressions of the cylindrical system. 5.34 State the mathematical expressions for the following operators and operations assum- ing a spherical coordinate system: nabla operator, Laplacian operator, gradient of a scalar, divergence of a vector, and curl of a vector. 5.35 Repeat the previous exercise for the general orthogonal coordinate system. 5.36 Express, in tensor notation, the mathematical condition for a vector field to be solenoidal. 5.37 Express, in tensor notation, the mathematical condition for a vector field to be irro- tational. 5.38 Express, in tensor notation, the divergence theorem for a differentiable vector field explaining all the symbols involved. Repeat the exercise for a differentiable tensor field of an arbitrary rank ( > 0 ). 5.39 Express, in tensor notation, Stokes theorem for a differentiable vector field explaining all the symbols involved. Repeat the exercise for a differentiable tensor field of an arbitrary rank ( > 0 ). 5.40 Express the following identities in tensor notation: ∇ · r = n ( a · r ) = a ∇ · ( ∇ × A ) = 0 ( fh ) = f h + h f
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5.7 Exercises 148 ∇ × ( f A ) = f ∇ × A + f × A A × ( B × C ) = B ( A · C ) - C ( A · B ) ∇ × ( ∇ × A ) = ( ∇ · A ) - ∇ 2 A ∇ · ( A × B ) = B · ( ∇ × A ) - A · ( ∇ × B ) ( A × B ) · ( C × D ) = A · C A · D B · C B · D 5.41 Prove the following identities using the language and techniques of tensor calculus: ∇ × r = 0 ∇ · ( f ) = 2 f ∇ × ( f ) = 0 ∇ · ( f A ) = f ∇ · A + A · ∇ f A · ( B × C ) = C · ( A × B ) = B · ( C × A ) A × ( ∇ × B ) = ( B ) · A - A · ∇ B ( A · B ) = A × ( ∇ × B ) + B × ( ∇ × A ) + ( A · ∇ ) B + ( B · ∇ ) A ∇ × ( A × B ) = ( B · ∇ ) A + ( ∇ · B ) A - ( ∇ · A ) B - ( A · ∇ ) B ( A × B ) × ( C × D ) = [ D · ( A × B )] C - [ C · ( A × B )] D
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Chapter 6 Metric Tensor The subject of the present chapter is the metric tensor which is one of the most important special tensors, if not the most important of all, in tensor calculus. Its versatile usage and functionalities permeate the whole discipline of tensor calculus.
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  • Summer '20
  • Rajendra Paramanik
  • Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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