1. Write the following using summation convention: (a) (x) + (x) + (x) (b) (x) + (x) + (x) + (c) af dx' af dx dx dt + df dr dx dt EXERCISE 49.1 +..... + af dx" dx" dt Mathematical Physics Ans. A = ' " Ans. -AP, B = klm df dt 2. Write out all the tensor in S = ar' x' taking n = 3. a. Ans. S = (ax + axx + at) + (axx + axx + axx) + (axx + 3. Write the tensor contained in x9, Xqr if n = 2 11 Ans. (x+x)x + (x + x)x2 + (x + x)x + (x + x)x (4. How many equations in a four dimensional space are represented by Rp = 0 Ans. 8. 5. Show that every tensor can be expressed as the sum of two tensors, one of which is symmetric and the other skew-symmetric in a pair of covariant or contravariant indices. Ans. (x) Ans. (x) 6. Show that the symmetric (or antisymmetric) property of a tensor is conserved under a transformation of co-ordinates. af dx' dx' di 7. If A and B, are components of a contravariant and covariant tensor of rank one, then show that C = A'B, are the components of a mixed tensor of rank 2. 8. Write down the laws of transformation for the tensors A and Bm 9. Evaluate (a) 88 (b) 0. Show that the velocity of a fluid at any point is a contravariant tensor of rank one. azxx + aztx) Thus we atisfy the ten Alternate tens the alternate t 49.22 KRO The sym dx ax ax dx dx' ' axm Drst Ans. (a) (b) 8 Above It mean In gene Exam Solut The