# On the left change i to s and j to n and on the right

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On the left, change i to s and j to n, and on the right change i to n and j to s, then e k ε snk x s x n φ k n s = e k ε snk x s x n φ k s n The order of summation does not matter, so e k ε snk x s x n φ k n s = 0 which shows that ∇ × ∇φ = 0 . 5. Show that a tensor plus its transpose, A + A t , is always a symmetric tensor.
6. Is the velocity field v that of an incompressible fluid if v = xe x + ze y ye z ?
so the flow is compressible. Ungraded exercise Write the following expressions, which are given in Gibbs notation, using summation notation. Include the unit vectors e . Label each as a scalar, vector or tensor. Gibbs notation B a A B A :B a × b ∇⋅ v v ( ) Solution Write the following expressions, which are given in Gibbs notation, using summation notation. Include the unit vectors. Label each expression as a scalar, vector or tensor. Gibbs notation Summation notation B a B ij a j e i vector A B A ij B jk e i e k tensor
a × b ε ijk a i b j e k vector ∇⋅ v v ( ) x i v i v j ( ) e j vector