This lecture introduces tensors and vectors through the following points: 1) Tensors are abstract mathematical entities that generalize the concept of vectors. Vectors are first-order tensors. 2) Coordinate systems like Cartesian and curvilinear coordinates are discussed. Vector components can be transformed between different coordinate systems. 3) Higher-order tensors are defined as real-valued multilinear functions of vectors, generalizing first-order tensors (vectors). Component transformations preserve the coordinate-invariant nature of tensors.

1. FACULTY: Asst Prof. Shaibal Sahoo
Lecture 1
Introduction to vectors and tensors
ENGINEERING MECHANICS

2. Lecture 1– Introduction to tensors and vectors
KINEMATICS OF DEFORMATIONS
2
deformed
configuration
thermo-mechanical loads
laws of nature .
CONSERVATION OF MASS
BALANCE OF LINEAR MOMENTUM
BALANCE OF ANGULAR MOMENTUM
LAWS OF THERMODYNAMICS
reference
configuration
CONTINUOUS
MEDIA
5 equations
16 unknown fields +
continuously varying fields
(time and space averages over
the underlying structure)
CONSTITUTIVE
EQUATIONS
11 equations
Empirical
observation
Tensor algebra
Tensor analysis
atomic/
micro/meso
structure
is revealed
Multi-scale
approaches
Experimental
mechanics and
thermodynamics

3. Lecture 1– Introduction to tensors and vectors
Review:
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DIY

4. What is a tensor?
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- Tensors are abstract mathematical entities.
- Vectors are first order tensors.
- Vectors and tensors exist separately of a particular
coordinate system (i.e., they are coordinate invariant).
… so let’s start by reviewing Vector Algebra.

5. Vector algebra
DIY
Which is the name of each property?
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Vector space

6. Vector algebra
Vector space – Euclidean Space
- In continuum mechanics, we restrict attention to
finite-dimensional spaces.
- We also need additional geometric properties,
such as distances and angles.
… but …
- What is an n-dimensional vector space?
- What is a distance?
- What is an angle?
Naturally, you have a good intuitive understanding of
these concepts, but we will formalize them with
the purpose of understanding their generalization.
Euclidean
Space
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7. Vector algebra
What is an n-dimensional vector space?
- Linear independent vectors
- Given a vector space, the largest possible number of
linearly-independent vectors is the dimensionality of
the vector space.
- Any set of
a basis of
linearly independent vectors can be selected as
.
- Every vector can be written as a unique linear combination of
the basis vectors.
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8. Vector algebra
What is a distance? What is an angle?
- Both can be defined from geometric interpretations of
an inner product and a norm.
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9. Vector algebra
What is a distance? What is an angle?
- Both can be defined from geometric interpretations of
an inner product and a norm.
- Euclidean space:
Euclidean
Space
DIY
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Let’s recall addition and multiplication in

10. Vector algebra
Euclidean
Space
What is a distance? What is an angle?
- Both can be defined from geometric interpretations of
an inner product and a norm.
- Euclidean space
- Euclidean point space
- Now we can naturally extend the notion of distance and angle to:
+ Distance between two points
+ Angle formed by three points
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11. Vector algebra
Coordinate systems
- An origin (relative to which positions are measured)
- A set of coordinate curves (that correspond to paths through space
along which all but one of the coordinates are constant)
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12. Vector algebra
Coordinate systems
- An origin (relative to which positions are measured)
- A set of coordinate curves (that correspond to paths through space
along which all but one of the coordinates are constant)
- Basis are defined at each position
in space as the tangent vectors
to the coordinate curves.
Therefore, basis vectors
change from position to position.
- Basis are non-orthogonal
in general.
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13. Vector algebra
DIY
What is the component of a vector (or the coordinate of a position vector)
along a basis vector direction?
Coordinate systems
- If the coordinate curves are straight lines then the system is a
rectilinear coordinate system.
- If the basis vectors of a rectilinear coordinate system are orthogonal
then the system is a Cartesian coordinate system.
- If the basis vectors (or axes) of a Cartesian coordinate system are
unit vectors then the basis are called orthonormal and
follow the the condition
- We will choose basis vectors that form a right-handed triad
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14. Vector algebra
Curvilinear coordinate systems
- Two set of basis vectors at each position in space
tangent vectors
reciprocal vectors
defined through
DIY
What is the component of a vector
along a basis vector direction?
- Examples:
Polar cylindrical coordinates
Spherical coordinates
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15. Vector algebra
Curvilinear coordinate systems
- Two set of basis vectors at each position in space
tangent vectors
reciprocal vectors
- Contravariant components
- Covariant components
- Connection between
components
with
defined through
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16. Vector algebra
Curvilinear coordinate systems
- All vector and tensor related operations (and continuum mechanics in
general) can be defined in curvilinear coordinate systems
- Example: the dot product
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Note:and are
components of
the metric tensor.
We will mostly limit ourselves to Cartesian coordinate systems

17. Vector algebra
Change of basis (… back to Cartesian systems)
- Two orthonormal bases and
- The goal is to change bases from
- A linear transformation matrix
to
is defined as
DIY
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Let’s write the transformation
using a column-matrix notation.
Show the following properties:

18. Vector algebra
Vector component transformation
- We opened this Lecture by saying that
Vectors and tensors are coordinate invariant, i.e., invariant with
respect to component transformation.
DIY
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Prove the following transformation rules

19. Vector algebra
Vector component transformation
- We opened this Lecture by saying that
Vectors and tensors are coordinate invariant, i.e., invariant with
respect to component transformation.
- A vector can be expressed as components in
Alternatively, it can be replaced by a linear mapping
.
which returns a real number equal to the projection of the vector
on . Therefore, when evaluated on orthonormal Cartesian basis,
it returns the coordinates of the vector
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20. DIY
From first order to n-th order tensors
- A first order tensor (vector) is a real-valued linear function of vectors
- An n-th order tensor is a real-valued n-linear function of vectors
- The components of a second-order tensor in a particular basis
are then defined as
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