By P. P. Dendy, R. Tuffnell, C. H. B. Mee
A booklet of questions, with labored options, from A-level, 6th shape access (Cambridge) and S-level papers.
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Additional info for Cambridge Problems in Physics and Advice on Solutions (2nd Edition)
Bn be arbitrary vectors. Then we can represent a dyadic as = A1 B1 + A2 B2 + · · · + An Bn . 1) The transpose of a dyadic is defined as the result obtained by the interchange of the two vectors in each of the dyads. For example, the transpose of the dyadic in Eq. 1) is T = B1 A1 + B2 A2 + · · · + Bn An . One of the properties of a dyadic is defined by the dot product with a vector, say V: · V = A1 (B1 · V) + A2 (B2 · V) + · · · + An (Bn · V), V· = (V · A1 )B1 + (V · A2 )B2 + · · · + (V · An )Bn .
A2n . A = .. .. , αA = ··· ··· ··· ··· . ... αam1 αam2 . . αamn am1 am2 . . amn Matrix addition has the following properties: 1. Addition is commutative: A + B = B + A. 2. Addition is associative: A + (B + C) = (A + B) + C. 3. There exists a unique matrix 0, such that A + 0 = 0 + A = A. The matrix 0 is called zero matrix; all elements of it are zeros. 4. For each matrix A, there exists a unique matrix −A such that A + (−A) = 0. 5. Addition is distributive with respect to scalar multiplication: α(A + B) = αA + αB.
2 Vector Algebra 21 3. ai = b j ci di . 4. xi xi = r 2 . 5. ai b j c j = 3. SOLUTION: 1. Not a valid expression because the free indices r and s do not match. 2. Valid; both m and s are free indices. There are nine equations (m, s = 1, 2, 3). 3. Not a valid expression because the free index j is not matched on both sides of the equality, and index i is a dummy index in one expression and a free index in the other; i cannot be used both as a free and dummy index in the same equation. The equation would have been valid if i on the left side of the equation is replaced with j; then there will be three equations.
Cambridge Problems in Physics and Advice on Solutions (2nd Edition) by P. P. Dendy, R. Tuffnell, C. H. B. Mee