Tensor Intoduction

2. • Tensorsare mathematical objects that generalize scalars,
vectors and matrices to higher dimensions.
• Ifyou are familiar with basic linear algebra, you should
have no trouble understanding what tensors are.
• Inshort, a single-dimensional tensor can be represented as a
vector.

3. • A tensor is a vector or matrix of n-dimensions that represents all
types of data.
• All values in a tensor hold identical data type with a known (or
partially known) shape.
• The shape of the data isthe dimensionality of the matrix or
array. A tensor can be originated from the input data or the
result of a computation.

4. • Tensors are simply mathematical objects that can
be used to describe physical properties, just like
scalars and vectors.
• In fact tensors are merely a generalisation of
scalars and vectors; a scalar is a zero rank tensor,
and a vector is a first rank tensor.

5. • A tensor isa generalization of vectors and matrices and is
easily understood as a multidimensional array.
• A tensor is a container which can house data in N dimensions.
Often and erroneously used interchangeably with the matrix
(which is specifically a 2-dimensional tensor), tensors are
generalizations of matrices to N-dimensional space.
Mathematically speaking, tensors are more than simply a
data container, however
.

6. WHAT IS A TENSOR?
• In Newtonian mechanics, we say that physical
quantities such as length, mass, energy, volume are
scalars while velocity, force, linear momentum,
electric field are vectors.

7. To answer this question recall the concept of scalars and vectors that you
have studied so far.
In school you learnt that scalar quantities have magnitude only, whereas
vector quantities have both magnitude and direction.
Now you know that the value of a scalar quantity does not change (or
remains invariant) on changing the coordinate system.
You have also learnt that a vector changes in a special way when it
undergoes a coordinate transformation.
Let us quickly revisit these concepts because these form the basis for
defining a tensor.

8.  Tensors, defined mathematically, are simply
arrays of numbers, or functions, that transform
according to certain rules under a change of
coordinates.
 multidimensional array of numbers.

9. n-DIMENSIONAL SPACE
• In three dimensional space a point is determined by a set of
three numbers called the co-ordinates of that point in
particular system.
• For example (x, y, z) are the co-ordinates of a point in
rectangular Cartesian co-ordinate system.

10. • By analogy if a point is respected by an ordered set of n
real variables (x1, x2, x3,……xi …..xn) or more
conveniently (𝑥1, 𝑥2, 𝑥3,….. 𝑥𝑖….. 𝑥𝑛 )
• Hence the suffixes 1, 2, 3, …, i, ….., n denote
variables and not the powers of the variables
involved], then all the points corresponding to all values
of co-ordinates (i.e., variables) are said to form an n-
dimensional space, denoted by Vn.

11. Tensors are important in physics because they provide a
concise mathematical framework for formulating and
solving physics problems in areas such as mechanics
(stress, elasticity, fluid mechanics, moment of inertia,
...),

12. scalar(rank 0 tensor)
vector(rank 1 tensor)
𝑣1
𝑣2
𝑣 3
⋮
𝑣𝑛
list of numbers extended in
downwards in one dimension.

13. Tensor of rank 0(SCALAR)
• A tensor may consist of a single number, in which
case it is referred to as a tensor of order zero, or
simply a scalar.
• For reasons which will become apparent, a scalar
may be thought of as an array of dimension zero
(same as the order of the tensor).

14. Tensor of rank 0- Example
• An example of a scalar would be the mass of a particle or
object.
• An example of a scalar field would be the density of a fluid as
a function of position.
• A second example of a scalar field would be the value of the
gravitational potential energy as a function of position.
• Note that both of these are single numbers (functions) that vary
continuously from point-to-point, thereby defining a scalar
field.

15. vector(rank 1 tensor)
𝑣1
𝑣2
𝑣 3
⋮
𝑣𝑛
list of numbers extended in
downwards in one dimension.so it is a one dimensional
array.

16. Matrix (Rank of 2 tensors)
rank two.
𝑚11 𝑚12…𝑚1𝑛
𝑚21
⋮
𝑚𝑛1
𝑚22…𝑚2𝑛
⋮
𝑚𝑛2 … 𝑚𝑛𝑛
Two dimensional grid(array) of numbers.

17. Rank 3 tensor
we can go higher and higher.
any multidimensional array of numbers is a tensor.

19. Tensor.
An object that is invariant under a change of
coordinates and has components that changes in a
special predictable way under the change of
coordinates.

20. • Mathematicians & engineers need a way to
geometrically represent physical quantities &
understand how they behave under different
coordinate systems.