1. Budge Budge Institute Of Technology
Name - Manish Kumar
Department-Mechanical Engineering
Sem-5th
University Roll No-2760072003
Paper Name - Solid Mechanics
Paper Code - (PC-ME 502)
Faculty Name-SKM
Topic- Introduction To Cartesian Tensors
2. A vector is a bookkeeping tool to keep track of two pieces of information (typically
magnitude and direction) for a physical quantity. Examples are position, force and velocity.
The vector has three components.
Vector qualities include:
1. displacement
2. 2. velocity
3. 3. acceleration
4. 4. force
5. 5. weight
6. 6. momentum
3. Types of Vectors
1. Zero Vector.
2. Unit Vector.
3. Position Vector.
4. Co-initial Vector.
5. Like and Unlike Vectors.
6. Co-planar Vector.
7. Collinear Vector.
8. Equal Vector.
Unit Vector:
A vector having unit magnitude is called a unit vector. A
unit vector in the direction of vector A is given by A unit
vector is unit less and dimensionless vector and
represents direction only
Unit Vector= Vector/vector’s magnitude
4. VV
Vector quantities change when:
their magnitude changes
their direction changes
their magnitude and direction both
Vector Algebra
Vectors algebra involves algebraic operations across vectors. The algebraic operations
involving the magnitude and direction of vectors is performed in vector algebra. Vector
algebra helps for numerous applications in physics, and engineering to perform addition
and multiplication operations across physical quantities, represented as vectors in three-
dimensional space.
Vectors carry a point A to point B. The length of the line between the two points A and B is
called the magnitude of the vector and the direction or the displacement of point A to
point B is called the direction of the vector AB. Vectors are also called Euclidean vectors or
Spatial vectors. Vectors have many applications in mathematics, physics, engineering, and
various other fields.
Vectors have an initial point at the point where they start and a terminal point that tells
the final position of the point. Various algebraic operations such as addition, subtraction,
and multiplication can be performed in vector algebra.
5. Representation of Vectors
Vectors are usually represented in bold lowercase such as a or using an arrow
over the letter as . can also be denoted by their initial and terminal points with
an arrow above them,
for example, vector AB can be denoted as . The standard form of representation
of a vectors is Here, a, b, c are real numbers and i, j, k are the unit vectors
along the x-axis, y-axis, and z-axis respectively. ai+bj +ck
Since the three components of A act in the positive i, j, and k directions, we can
write A in Cartesian vector form as
6. Co-Initial Vectors
A vector is said to be a co-initial vector when two or more
vectors have the same starting point, for example, Vectors AB
and AC are called co-initial vectors because they have the same
starting point A.
Like and Unlike Vectors
The vectors having the same directions are said to be
like vectors whereas vectors having opposite directions are said
to be unlike vectors.
Coplanar Vectors
Three or more vectors lying in the same plane are known as
coplanar vector
Collinear Vectors
Vectors that lie in the parallel line or the same line concerning
their magnitude and direction are known to be collinear vectors,
also known as parallel vectors.
Equal Vectors
Two vectors are said to be equal vectors when they have both
direction and magnitude equal, even if they have different initial
points. Displacement Vector
The vector AB represents a displacement vector if a point is
displaced from position A to B.
7. Negative of a Vector
Suppose a vector is given with the same magnitude and direction, now if
any vector with the same magnitude but the opposite direction is given
then this vector is said to be negative of that vector. Consider two vectors a
and b, such that they have the same magnitude but opposite in direction
then these vectors can be written as a = – b
Orthogonal Vectors:
Two or more vectors in space are said to be orthogonal if the angle
between them is 90 degrees. In other words, the dot product of orthogonal
vectors is always 0.
Free vector
A vector that can be displaced parallel to itself and applied at any point is
called a free vector.
Null vector
A vector whose magnitude is zero and has no direction, it may have all
directions is said to be a null vector.
A null vector can be obtained by adding two or more vectors
8. Intoduction To Cartisian Tensor
Tensor
A tensor is a quantity, for example a stress or a strain, which has magnitude,
direction and a plane in which it acts.
Stress and strain are both tensor quantities.In real engineering components,
stress and strain are 3-D tensors.
Whenever we want to represent some physical quantity mathematically, wneed
to see how much information is needed to specify the value of that quantity.
▶ 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
▶ The need for second rank tensors comes when we need to consider more than
one direction to describe one of these physical properties.
9. ▶ 1. Rank 0 Tensor:
The familiar scalar is the simplest tensor and is a rank 0 tensor. Scalars are just
single real numbers like ½, 99 or -1002 that are used to measure magnitude (size).
Scalars can technically be written as a one-unit array: [½], [99] or [-1002], but it’s
not usual practice to do so. •
Scalar:
Tensor of rank 0. (magnitude only – 1 component)
2. Rank 1 Tensor:
Vectors are rank 1 tensors. There are many ways to write vectors, including as an
array:
3. Rank 2 Tensor:
The next level up is a Rank 2 tensor, which can be represented by a matrix.
Matrices are rectangular arrays of numbers arranged into columns and rows (similar
to a spreadsheet). They have a rank of 2 because of the two-dimensional array •
Vector: Tensor of rank 1. (magnitude and one direction – 3 components) •
Dyad:
Tensor of rank 2. (magnitude and two directions – 3 2 = 9 components) •
Triad:
Tensor of rank 3. (magnitude and three directions – 3 3 = 27 components)
