1. Write the following using summation convention: (a) (x) + (x) + (x) (b) (x) +...

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1. Write the following using summation convention: (a) (x¹)¹ + (x¹)² + (x¹)³ (b) (x¹)² + (x)² + (x³)² + (c) af dx' af dx² dx² dt + df dr dx¹ dt EXERCISE 49.1 +..... + af dx" dx" dt Mathematical Physics Ans. A = ах' ах ах" ахр аху ахк Ans. -AP, B = klm df dt 2. Write out all the tensor in S = ar' x' taking n = 3. a. Ans. S = (a₁¹x¹ + a₁₂x¹x² + a₁t¹²³) + (a₂₁x²x¹ + a₂₂x²x² + a₂²x²x³) + (a₁₁x²x¹ + 3. Write the tensor contained in x9, Xqr if n = 2 11 Ans. (x¹+x²¹)x₁ + (x¹¹ + x²¹)x₁2 + (x¹² + x²²)x₂ ₁ + (x¹² + x²²)x₂² (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. (x¹) Ans. (x¹)² 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 A¹ 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. azx²³x² + a²zt³x³) Thus we atisfy the ten Alternate tens the alternate t 49.22 KRO The sym dx ax ax dx³ dx' ахр аху алк ах' axm Drst Ans. (a) (b) 8 Above It mean In gene Exam Solut The 1. Write the following using summation convention: (a) (x¹)¹ + (x¹)² + (x¹)³ (b) (x¹)² + (x)² + (x³)² + (c) af dx' af dx² dx² dt + df dr dx¹ dt EXERCISE 49.1 +..... + af dx" dx" dt Mathematical Physics Ans. A = ах' ах ах" ахр аху ахк Ans. -AP, B = klm df dt 2. Write out all the tensor in S = ar' x' taking n = 3. a. Ans. S = (a₁¹x¹ + a₁₂x¹x² + a₁t¹²³) + (a₂₁x²x¹ + a₂₂x²x² + a₂²x²x³) + (a₁₁x²x¹ + 3. Write the tensor contained in x9, Xqr if n = 2 11 Ans. (x¹+x²¹)x₁ + (x¹¹ + x²¹)x₁2 + (x¹² + x²²)x₂ ₁ + (x¹² + x²²)x₂² (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. (x¹) Ans. (x¹)² 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 A¹ 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. azx²³x² + a²zt³x³) Thus we atisfy the ten Alternate tens the alternate t 49.22 KRO The sym dx ax ax dx³ dx' ахр аху алк ах' axm Drst Ans. (a) (b) 8 Above It mean In gene Exam Solut The

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