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Rai Saheb Bhanwar Singh College Nasrullaganj
Rai Saheb Bhanwar Singh College Nasrullaganj

Tensors obey algebraic properties including addition, multiplication, contraction, and symmetrization. Addition of tensors combines their components. Multiplication of tensors combines their indices and ranks to form a new tensor. Contraction sets equal a covariant and contravariant index, reducing the tensor's rank. Symmetric tensors do not change sign under index interchange, while antisymmetric tensors change sign.

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NPTEL Physics Mathematical Physics - 1 Lecture 28 Algelraic properties of tensors Like vectors, the tensors obey certain operations which are: a) Addition If T and S are two tensors of type (r, s), then their sum U = T + S is defined as 1 1.. = 1.. + 1.. 1 1 Thus U is also a tensor of type (r, s), which can be easily proved by showing the transformation property, 1.. = ヌ 1 モ1 1 ヌ モ ヌ 1 ヌ 1 ヰ 1.. Page 13 of 20 Joint initiative of IITs and IISc Funded by MHRD 1 b) Multiplication If T is a tensor of type (1, 1) and S is a tensor of type (2, 2), then the product U = T S, defined component wise as, 1 1+2 1 1 1+2 1.. 1+ 2 = 1.. 1 1+1 .. 1+ 2 1 +1 Which is a tensor of type (1 + 2, 1 + 2). For example, if T is a tensor of type (1,2) with components and S is a tensor type (2,1) with components , then the components of the tensor product U as, and they transform according to the rules, = = = The above shows that U is a tensor of rank (3, 3). c) Contraction The contraction is defined by the following operation given by a tensor type (r, s), take a covariant index and set it equal to a contravariant index, that is, sum over those two indices. It will result in a tensor of type (r-1, s-1). An example will make it clear. Take a tensor of type (2, 1) whose ヌ 2 ヌ ヌ ヰ ヌ ヌ2 ヰ ヰ ヰ
Page 14 of 20 Joint initiative of IITs and IISc Funded by MHRD NPTEL Physics Mathematical Physics - 1 components are and set k = j. Now how do the components of transform? = ヌ ヌ = ヌ ヰ ヰ ヌ モ This shows that transforms as components of a contravariant tensor of type (1, 0). Of specific interest is a tensor of type (1, 1). Contracting this, one will get a tensor of type (0, 0) i.e. a scalar. Let 癌 is a contravariant vector with components 基 and 汲 is a covariant vector with components 巨. Then = 巨 is a tensor of type (1, 1). When one contracts it, one gets = A Bi which is a scalar as we have taken dot product of two vectors. i Symmetrization Some of the tensors we come across in physics have the property that when two of their indices are interchanged, the tensors either change or do not change sign. The ones which do not change sign are called as symmetric tensors and those which change sign under change of indices are called as antisymmetric tensors. Examples are = ji ; is a symmetric tensor = ji ; is an antisymmetric tensor

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lec28.ppt

  • 1. NPTEL Physics Mathematical Physics - 1 Lecture 28 Algelraic properties of tensors Like vectors, the tensors obey certain operations which are: a) Addition If T and S are two tensors of type (r, s), then their sum U = T + S is defined as 1 1.. = 1.. + 1.. 1 1 Thus U is also a tensor of type (r, s), which can be easily proved by showing the transformation property, 1.. = ヌ 1 モ1 1 ヌ モ ヌ 1 ヌ 1 ヰ 1.. Page 13 of 20 Joint initiative of IITs and IISc Funded by MHRD 1 b) Multiplication If T is a tensor of type (1, 1) and S is a tensor of type (2, 2), then the product U = T S, defined component wise as, 1 1+2 1 1 1+2 1.. 1+ 2 = 1.. 1 1+1 .. 1+ 2 1 +1 Which is a tensor of type (1 + 2, 1 + 2). For example, if T is a tensor of type (1,2) with components and S is a tensor type (2,1) with components , then the components of the tensor product U as, and they transform according to the rules, = = = The above shows that U is a tensor of rank (3, 3). c) Contraction The contraction is defined by the following operation given by a tensor type (r, s), take a covariant index and set it equal to a contravariant index, that is, sum over those two indices. It will result in a tensor of type (r-1, s-1). An example will make it clear. Take a tensor of type (2, 1) whose ヌ 2 ヌ ヌ ヰ ヌ ヌ2 ヰ ヰ ヰ
  • 2. Page 14 of 20 Joint initiative of IITs and IISc Funded by MHRD NPTEL Physics Mathematical Physics - 1 components are and set k = j. Now how do the components of transform? = ヌ ヌ = ヌ ヰ ヰ ヌ モ This shows that transforms as components of a contravariant tensor of type (1, 0). Of specific interest is a tensor of type (1, 1). Contracting this, one will get a tensor of type (0, 0) i.e. a scalar. Let 癌 is a contravariant vector with components 基 and 汲 is a covariant vector with components 巨. Then = 巨 is a tensor of type (1, 1). When one contracts it, one gets = A Bi which is a scalar as we have taken dot product of two vectors. i Symmetrization Some of the tensors we come across in physics have the property that when two of their indices are interchanged, the tensors either change or do not change sign. The ones which do not change sign are called as symmetric tensors and those which change sign under change of indices are called as antisymmetric tensors. Examples are = ji ; is a symmetric tensor = ji ; is an antisymmetric tensor