Page 32 - Intro to Tensor Calculus

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EXERCISE 1.1

I 1. Simplify each of the following by employing the summation property of the Kronecker delta. Perform
sums on the summation indices only if your are unsure of the result.

(a) e ijk δ kn (c) e ijk δ is δ jm δ kn (e) δ ij δ jn
(d)
(b) e ijk δ is δ jm a ij δ in (f) δ ij δ jn δ ni

I 2. Simplify and perform the indicated summations over the range 1, 2, 3

(a) δ ii (c) e ijk A i A j A k (e) e ijk δ jk
(b) δ ij δ ij (d) e ijk e ijk (f) A i B j δ ji − B m A n δ mn

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I 3. Express each of the following in index notation. Be careful of the notation you use. Note that A = A i
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is an incorrect notation because a vector can not equal a scalar. The notation A · b e i = A i should be used to
express the ith component of a vector.

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(a) A · (B × C) (c) B(A · C)
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(b) A × (B × C) (d) B(A · C) − C(A · B)
I 4. Show the e permutation symbol satisfies: (a) e ijk = e jki = e kij (b) e ijk = −e jik = −e ikj = −e kji
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I 5. Use index notation to verify the vector identity A × (B × C)= B(A · C) − C(A · B)
I 6. Let y i = a ij x j and x m = a im z i where the range of the indices is 1, 2
(a)Solve for y i in terms of z i using the indicial notation and check your result
to be sure that no index repeats itself more than twice.
(b) Perform the indicated summations and write out expressions
for y 1 ,y 2 in terms of z 1 ,z 2
(c) Express the above equations in matrix form. Expand the matrix

equations and check the solution obtained in part (b).

I 7. Use the e − δ identity to simplify (a) e ijk e jik (b) e ijk e jki

I 8. Prove the following vector identities:
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(a) A · (B × C)= B · (C × A)= C · (A × B) triple scalar product
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(b)(A × B) × C = B(A · C) − A(B · C)
I 9. Prove the following vector identities:
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(a)(A × B) · (C × D)= (A · C)(B · D) − (A · D)(B · C)
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(b) A × (B × C)+ B × (C × A)+ C × (A × B)= 0
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(c) (A × B) × (C × D)= B(A · C × D) − A(B · C × D)