What is Vector ?

What is Tensor ?
• Tensors are merely a generalization of scalars and vectors. A tensor is a multi-dimensional array of numbers.
Array means a grid of numbers.
• The rank (or order) of a tensor is defined by the number of directions (and hence the dimensionality of the
array) required to describe it.
Tensor is quantity which depends upon magnitude,
direction and plane.
o Scalar: Tensor of rank 0 (Magnitude only)
o Vector: Tensor of rank 1 (Magnitude and one direction)
o Dyad: Tensor of rank 2 (Magnitude and two directions)
o Triad: Tensor of rank 3 (Magnitude and three directions)

σxy = Stress on the x

surface in y direction
 Summation can be droped
in Einstein’s convention

n is dimensionality of space we have two different symbol groups here one

group is a aijxj and another group is bi. The terms
which are written as a product or written as
A free index means an "independent
independent are called a symbol group.
dimension" or an order of the tensor
whereas a dummy index means if some index is being repeated in a symbol
summation. group then that index is called a dummy index
 Can’t replace
dummy index
with free
index

J is counted as 2 and is dummy index

Here i is the free index (i appears once), k appears twice, j

appears twice, hence k and j are dummy index.
Types of Stress

Stress

Normal Stress Shear Stress

Tensile Stress

Compressive Stress
1.1.1 Tensile Stress
It is induced in a body when it is subjected to two equal and opposite
pulls and is called tensile stress.
Effect of Tensile Stress

• Due to the tensile stress there is an increase in the

length of the body and decrease in the cross section
area of the body.

• Tensile stress is a type of normal stress, so it acts at

90 degree to the area.
1.1.2 Compressive Stress
• The stress which induced in a body when
it is subjected to two equal and opposite pushes.
Effect of Compressive Stress
•Due to the compressive stress, there is a decrease in
the length and increase in the cross section area of the
body.

•Compressive stress is also a type of normal stress and

so it also acts at 90 degree to the area
 1.2 Shear stress

• It is induced in a body when it is

subjected to two equal and opposite forces that acts tangential to the
area.

• It acts in plane to the stressed area at right-angles to compressive or

tensile stress.
• Stress parallel to a plane is shear stress and can be
expressed as
Effect of Shear Stress
•Shear stress is responsible for the
change in the shape of the body. It does on affect the
volume of the body.

•It is developed when external loads tend to cause the

two segments of a body to slide over one another.
2.2 On the basis of Type of loading
2.2.1 Normal Strain
Normal Strain

Compressive strain Tensile Strain

• Compressive Strain
It is strain measure under compressive loading.
It tends to increase the cross section area and decrease
the length of the material

An elastic wire
• Tensile Strain
It is strain measure under tensile loading.
It tends to decrease the cross section area and increase
the length of the material

An elastic wire
2.2.2 Shear Strain
When the shear stress is applied on a material, it
tends to deform the shape of the material. The
change in tangential angle in the direction of the
loading is called shear strain.
2.2.3 Volumetric strain
The ratio of change in volume to original volume under
normal loading condition is known as volumetric strain.

Volumetric 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑉𝑜𝑙𝑢𝑚𝑒

=
strain 𝑂𝑟𝑖𝑔𝑛𝑎𝑙 𝑉𝑜𝑙𝑢𝑚𝑒
Along with the normal stresses, σxx = Stress on the x
tangential stress can be applied. surface in the x direction

Axis/surface Direction of Force σxx

σxy

x surface

σxy = Stress on the x

surface in y direction
Application
• One can get a lot of insight into how geometry of complex systems work

Geometry at elementary level How space-time is curved ?

The Einstein field equations (EFE; also known
as Einstein's equations) comprise the set of 10
equations in Albert Einstein's general theory of
relativity that describe the fundamental
interaction of gravitation as a result of
spacetime being curved by mass and energy.
The Einstein field equations (EFE) may be
written in the form:

where Rμν is the Ricci curvature tensor, R is the

scalar curvature, gμν is the metric tensor, Λ is
the cosmological constant, G is Newton's
gravitational constant, c is the speed of light in
vacuum, and Tμν is the stress–energy tensor.

Source: wikipedia
• Superposition of quantum states
It is same as that of linear combination so with linear combinations. As simpler
vectors are combine together using scaling constants and addition to get a more
complicated vector. It turns out in quantum mechanics physical states are actually
just vectors (or tensors) and so we can combine simpler states together using linear
combinations to give us more complicated states

• Quantum entanglement
when two quantum systems are entangled together what we really mean is that
these state vectors have been combined together using something called the
tensor product

Tensor product symbol: it takes the geometrical space where

the first system lives and the geometrical space where the
second system lives and combines them together to create a
more complicated geometrical space (that's where the
entangled system lives).
Electrical Conductivity

A current density and an electric field are established in a

conductor whenever a potential difference is maintained across
the conductor.
Vectors and Tensors
When vectors are in
the same plane but
are not acting along
the same line of action
they are known as
non-collinear vectors.
Covariant and Contravariant Components of
Vectors
• In tensor analysis, covariant and contravariant components of vectors describe how the vector changes
geometrically with a change of basis.
• In example below, the basis vectors grew by a factor of 2, the vector components shrunk by a factor of 2

Below basis vectors are rotated in clockwise and the components rotated
counterclockwise
• Vectors exhibit this behavior of changing scale inversely to changes in scale to the
reference axes: they are contravariant.

• In contrast, the components of a convector change in the same way as changes to

scale of the reference axes: they are covariant. An example of a covector is the
gradient, which has units of a spatial derivative.
Invariance of Tensors
• Tensors and the vectors are invariant under coordinate
transformations (remain unchanged) but their components vary with
the variation in the reference axis (basis).
superscript Both indices are
opposite
 Displacement vector which is contravariant vector

New basis vectors

Einstein’s summation convention

j is same

To convert a vector from old basis to new one. One need to multiply old vector with a factor (which is
in this case is rate of change of new basis w.r.t old basis. Note that summation sign is removed as per
Einstein’s summation convention, so when i=1 then j will be 2 and 3 for 3D case.
Әx’i/Әxj = how much of xi is along the xj component or it is just the direction cosine
(cosine of the angle between the xi and xj system)
 Consider Gradient
of a scalar field

scalar remain totally invariant

Displacement is contravariant,
under coordinate
so should be written as
transformation, therefore scalar
superscript from here on.
remains same before and after
the transformation.
Invariance of Tensors
• How to move back and forth between different basis ? Word basis just means a set of
vectors that give us the main coordinate directions for our space. Tensors are invariant
under a coordinate system change so understanding how to move back and forth
between coordinate systems is really important for understanding tensors.

Forward Transformation
Backward Transformation

Storing above four

numbers in a 2 by 2 matrix
 Multiplying two matrices

Note: Vectors are invariant but vector components are not invariant
Forward Transformation in n Dimensions

To build the new basis vectors out of the old basis vectors and we just need to choose
the right coefficients

Backward Transformation in n Dimensions

• forward and backward transformations are inverses of each other because when we multiply the F and B
matrices we get the identity matrix, whose entries are given by the Kronecker Delta symbol

Kronecker Delta provides entries of the identity matrix