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Lecture notes on introduction to tensors

K. M. Udayanandan Associate Professor Department of Physics Nehru Arts and Science College, Kanhangad

Syllabus Tensor analysis-Introduction-definition-definition of different rank tensors-Contraction and direct product-quotient rule-pseudo tensors- General tensors-Metric tensors

• 1 Introducing Tensors
  • 1.1 Scalars or Vectors?
  • 1.2 Vector Division
  • 1.3 Moment of inertia
• 2 Re defining scalars and vectors
  • 2.1 Cartesian Tensors
    • 2.1.1 Scalars
    • 2.1.2 Vectors
    • 2.1.3 Tensors
    • 2.1.4 Summation Convention
• 3 Quotient Rule
• 4 Non-Cartesian Tensors-Metric Tensors
  • 4.1 Spherical Polar Co-ordinate System
  • 4.2 Cylindrical coordinate system
• 5 Algebraic Operation of Tensors - 5.0.1 Definition of Contravariant and Co variant vector - 5.0.2 Exercises - 5.0.3 Co variant vector
  • 5.1 Addition & Subtraction of Tensors
  • 5.2 Symmetric and Anti symmetric Tensors
  • 5.3 Problems
  • 5.4 Contraction, Outer Product or Direct Product
• 6 Pseudo Scalars and Pseudo Vectors and Pseudo Tensors
  • 6.1 Pseudo Vectors
  • 6.2 Pseudo scalars
  • 6.3 General Definition
  • 6.4 Pseudo Tensor
• 7 University Questions-Solved

Chapter 1

Introducing Tensors

In our daily life we see large number of physical quantities. Tensor is the mathematical tool used to express these physical quantities. Any physi- cal property that can be quantified is called a physical quantity. The important property of a physical quantity is that it can be measured and expressed in terms of a mathematical quantity like number. For example, ”length” is a physical quantity that can be expressed by stating a number of some basic measurement unit such as meters, while ”anger” is a property that is difficult to describe with a number. Hence we will not call ’anger’ or ’happiness’ as a physical quantity. The physical quantities so far identified in physics are given below. Carefully read them. While reading observe that some are expressed unbold, some are bold fonted and some are large and bold. The known physical quantities are absorbed dose rate, acceleration, angular acceleration, angular speed, angular momentum, area, area density, capacitance, catalytic activity, chemical potential, molar concentra- tion, current density, dynamic viscosity, electric charge, electric charge

permeability, permittivity sometimes behave as scalars and some times not. The above mentioned physical quantities like mass, susceptibility. moment of inertia, permeability and permittivity obey very familiar equations like.

F^ ~ = m~a, P~ = χ E,~ L~ = I~ω, B~ = μ H, ~ D~ =  E, ~ F~ = T A, ~ J~ = σ E~

from which we can write m =

F~

~a χ =

P~

E~

I =

L~

~ω

etc. Consider the last case. Let L~ = 5ˆi+5ˆj +5ˆk and ~ω = ˆi+ˆj + ˆk. If you find moment of inertia in this case you will get it as 5. But if ~L = 9ˆi + 4ˆj + 11ˆk and ω is not changed we will not be able to divide ~L with ~ω and get the moment of inertia. Why this happen? What mathematical quantity is mass, susceptibility or moment of inertia? To understand this we must have a look at the concept of vector division once again.

1.2 Vector Division

Consider a ball thrown vertically downwards into water with a velocity

~v = 6ˆk

After entering water the velocity is decreased but the direction may not change. Then the new velocity may be

~v′^ = 3ˆk = 0. 5 ~v

Thus we transform the old velocity to a new velocity by a scalar multiple. But this is not true in all cases. Suppose the ball is thrown at an angle then the incident velocity may look like

~v = 5ˆi + 6ˆj + 8ˆk

and the deviated ball in the water may have different possible velocity like

~v′^ = 3ˆi + 2ˆj + 5ˆk ~v′^ = 2ˆi + 6ˆj + 7ˆk

etc. Consider the first case. The components of the final vector (3,2,5) can be obtained in different ways. Among them some are given below.

  

The above transformation matrix is diagonal.   

1.3 Moment of inertia

Finding the components of moment of inertia is the simplest example given in many textbooks introducing nine component physical quantity. We repeat it here for the simplicity and also for students who may be new at such derivations. Consider , L^ ~ = I~ω

In terms of ~r and ~p L~ = ~r × ~p = ~r × m~v = ~r × m (~ω × ~r) = m~r × (~ω × ~r)

We’ve A^ ~ ×^ (^ B~ × C~^ ) =^ (^ A. ~C~^ )^ B~ −^ (^ A. ~B~^ )^ C~ ∴ m~r × (~ω × ~r) = m (~r.~r) ~ω − m~r (~r.~ω) = mr^2 ~ω − m~r (~r.~ω)

= m

[(

x^2 + y^2 + z^2 )

ωxˆi + ωyˆj + ωz kˆ

)]

−m

[(

xˆi + yˆj + zˆk

(xωx + yωy + zωz )

]

then the three components of L~ are as follows,

Lx = m (y^2 + z^2 )^ ωx − mxyωy − mxzωz

Ly = −myxωx + m (x^2 + z^2 )^ ωy − myzωz Lz = −mzxωx − mzyωy + m (x^2 + y^2 )^ ωz

Lx = Ixxωx + Ixyωy + Ixz ωz Ly = Iyxωx + Iyyωy + Iyz ωz Lz = Izxωx + Izyωy + Izz ωz

where Ixx = m ((x^2 + y^2 + z^2 )^ − x^2 )^ = m (y^2 + z^2 ) Iyy = m ((x^2 + y^2 + z^2 )^ − y^2 )^ = m (x^2 + z^2 ) Izz = m ((x^2 + y^2 + z^2 )^ − z^2 )^ = m (x^2 + y^2 ) Ixy = −mxy = Iyz Iyz = −myz = Izy Izx = −mzx = Ixz

Thus ~L = I~ω can be written in the matrix form as

  

Lx Ly Lz

Ixx Ixy Ixz Iyx Iyy Iyz Izx Izy Izz

ωx ωy ωz

Thus I is a physical quantity with nine components.

I =

Ixx Ixy Ixz Iyx Iyy Iyz Izx Izy Izz

Chapter 2

Re defining scalars and vectors

We can see that scalars have one(3^0 ) component, vectors have 3 (3^1 ) com- ponets and tensors we saw had 9 (3^2 ) components. This shows that all these mathematical quantities belong to a family in 3 dimensional world. Hence these three types of physical quantities must have a common type of defi- nition. Since we classify them based on components we will redefine them based on componets. Hence we can redefine the scalars and vectors using coordinate transformation of components.

If the coordinate transformation is from cartesian to cartesian we call all the quantities as cartesian tensors and if the transformation is from carte- sian to spherical polar or cylindrical we call them as non cartesian tensors. First we will study cartesian tensors.

2.1 Cartesian Tensors

2.1.1 Scalars

Under co-ordinate transformation, a scalar quantity has no change. ie When we measure a scalar from a Cartesian or a rotated cartesian coordinate system the value of scalar remains invariant Hence

S′^ = S

For if we measure mass of some substance say sugar standing straight and then slightly tilting we will get the same mass. Thus any invariant quantity under coordinate transformation is defined as a scalar.

2.1.2 Vectors

Consider a transformation from Cartesian to Cartesian coordinate systems. Consider X 1 &X 2 represent the unrotated co-ordinate system and X 1 ′&X 2 ′ represent the rotated co-ordinate system. Let r be a vector and φ be the angle between x 1 axis and ~r. Then x 1 = r cos φ and x 2 = r sin φ. Let the coordinates be rotated through an angle θ. Then

x′ 1 = r cos (φ − θ) = r (cos φ cos θ + sin φ sin θ)

x′ 2 = r sin (φ − θ) = r (sin φ cos θ − cos φ sin θ) x′ 1 = x 1 cos θ + x 2 sin θ

∂x′ 1 ∂x 1 =^ a^11

∂x′ 1 ∂x 2 =^ a^12 ∂x′ 2 ∂x 1 =^ a^21

∂x′ 2 ∂x 2 =^ a^22 Thus the transformation equation can also be written as

x′ i =

j

∂x′ i ∂xj^ xj

This is also the transformation equation for a vector. Let us proceed and find the transformation from primed to unprimed coordinate system. We know that for transformation from from X to X’   x

′ 1 x′ 2

 cos^ θ^ sin^ θ − sin θ cos θ

 x^1 x 2

 x

′ 1 x′ 2

 = A

 x^1 x 2

where

A =

 cos^ θ^ sin^ θ − sin θ cos θ

Now AT^ =

 cos^ θ^ −^ sin^ θ sin θ cos θ

We can show that AAT^ = I = AA−^1

which means A−^1 = AT

A−^1

 x

′ 1 x′ 2

 = A−^1 A

 x^1 x 2

Rearranging (^) 

 x^1 x 2

 = A−^1

 x

′ 1 x′ 2

 x^1 x 2

 cos^ θ^ −^ sin^ θ sin θ cos θ

 x

′ 1 x′ 2

x 1 = ∂x ∂x^1 ′ 1 x′ 1 + ∂x ∂x^1 ′ 2 x′ 2

x 2 = ∂x ∂x^2 ′ 1

x′ 1 + ∂x ∂x^2 ′ 2

x′ 2

Then xi =

j

ajix′ j

Thus we can conclude and say that a vector is a physical quantity which transform like

A′ i =

j

aij Aj

where transformation is from primed coordinate system to un primed co- ordinate system. Now if the transformation is from primed to unprimed coordinate system the equation is

Now using the inverse transformation equation for Eq,

σ′ ij E j′ = aip σpq ajqE′ j

Then we get σ′ ij = aip ajqσpq

Thus we can see that a tensor of rank 2 will have 2 coefficients or the rank of a tensor can be obtained from counting the number of coefficients.

Thus we can define tensors in general as

A′ = A

tensor of rank zero or scalar

A′ i =

j

aij Aj

is tensor of rank one or vector and

A′ ij =

p

q

aip ajqApq

is a tensor of rank 2 and if have 3 coefficients we will get tensor of rank 3 etc.

2.1.4 Summation Convention

In writing an expression such as a 1 x^1 + a 2 x^2 + ..... + aN xN^ we can use the short notation

∑N

j=1^ aj^ x

j (^). An even shorter notation is simply to write it as

aj xj^ , where we adopted the convention that whenever an index (subscript or superscript)is repeated in a given term we are to sum over that index from 1 to N unless otherwise specified. This is called the “summation conven- tion. Any index which is repeated in a given term ,so that the summation convention applies , is called dummy index or umbral index. An index occurring only once in a given term is called a free index. In tensor analy- sis it is customary to adopt a summation convention and subsequent tensor equations in a more compact form. As long as we are distinguishing between contravariance and covariance, let us agree that when an index appears on one side of an equation, once as a superscript and once as a subscript (except for the coordinates where both are subscripts), we automatically sum over that index.