ﺗﺤﻠﯿﻞ اﻟﻤﺘﺠﮭﺎت
Vector Analysis
Level(3): phy
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lecture 1
Vectors and Scalars
Algebra of vectors
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Course Description:
1. Vectors and Scalars
2. Algebra of vectors,
3.Vector differentiation and vector integration,
4.Gradient, Divergence and Curl.
5.Physical significance of Gradient, Divergence and
Curl.
.
3.
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5. Green theorem,
6. Divergence theorem,
7.Gauss and Stoke’s theorem and their
applications
8. and curvilinear coordinate systems..
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وﺻﻒ اﻟﻤﻘﺮر:
❖
❖
❖
❖
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ﺗﺤﻠﯿﻞ اﻟﻤﺘﺠﮭﺎت :اﻟﻤﺘﺠﮭﺎت
واﻟﻘﯿﺎﺳﺎت،
ﺟﺒﺮ اﻟﻤﺘﺠﮭﺎت،
ﺗﻔﺎﺿﻞ اﻟﻤﺘﺠﮭﺎت وﺗﻜﺎﻣﻞ
اﻟﻤﺘﺠﮭﺎت،
اﻟﺘﺪرج ،اﻟﺘﺒﺎﻋﺪ ،واﻟﺘﻔﺎف
❖ اﻷھﻤﯿﺔ اﻟﻔﯿﺰﯾﺎﺋﯿﺔ ﻟﻠﺘﺪرج ،اﻟﺘﺒﺎﻋﺪ،
واﻟﺘﻔﺎف
❖ ﻧﻈﺮﯾﺔ ﻏﺮﯾﻦ ،ﻧﻈﺮﯾﺔ اﻟﺘﺒﺎﻋﺪ،
ﻣﺒﺮھﻨﺔ ﻏﺎوس وﺳﺘﻮك وﺗﻄﺒﯿﻘﺎﺗﮭﻤﺎ،
❖ أﻧﻈﻤﺔ اﻹﺣﺪاﺛﯿﺎت اﻟﻤﻨﺤﻨﯿﺔ
Vectors and Scalars:
Introduction:
In this chapter we shall use the ideas of the plane to develop a
new
mathematical concept, vector. If you have studied physics, you
have
encountered this concept in that part of physics concerned with
forces and
equilibrium
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the mathematical concept of vectors has become
important in
its own right and has extremely wide application, not
only in the sciences
but in mathematics as well.
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Scalar quantity
A ِquantity which is completely specified by a
certain numberِِ associated with a suitable unit
without any mention of direction in space is
known as scalar. Examples of scalar are time,
mass, length, volume, density, temperature,
energy, distance, speed etc
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Vector quantity:
A quantity which is completely described only
when both their
magnitude and direction are specified is known as
vector. Examples of
vector are force, velocity, acceleration,
displacement, torque, momentum,
gravitational force, electric and magnetic
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Vector Representations:
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Types of Vectors:
❖ Unit Vector: A vector whose
magnitude is unity
❖ Free Vector:A vector whose
position is not fixed in space. Thus,
the line of action of a free vector
can be shifted parallel to itself
❖ Localized or Bounded
Vectors:A vector which
cannot be shifted parallel
to itself
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❖ Coplanar Vectors:The
vectors which lies in the
same plane are called
coplanar vectors,
Null or Zero Vector: It is a vector
whose magnitude is zero. We denote the
null vector by O. The direction of a zero
vector is arbitrary. The vectors other
than zero vectors are proper vectors or
non-zero vectors.
Equal Vectors:
Parallel and Collinear Vectors:
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Concurrent Vectors : The vectors
which pass through the common point Negative of a Vector:
are called concurrent vectors
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Subtraction of Vectors:
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Addition and Subtraction of Vectors:
Addition of Vectors:
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/2اﻟﻀﺮب اﻟﻘﯿﺎﺳﻲ :
/3اﻟﻀﺮب اﻟﻨﻘﻄﻲ:
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/3اﻟﻀﺮب اﻟﺘﻘﺎطﻌﻲ :
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:
Example :-
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H.W
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ﺧﻮاص اﻟﻤﺘﺠﮭﺎت
/1ﺟﻣﻊ اﻟﻣﺗﺟﮭﺎت:
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اﻟﻤﺘﺠﮭﺎت اﻟﻤﺮﻛﺒﺔ:
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ﺛﻮاﺑﺖ
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lecture 2
Vectors and Scalars
Algebra of vectors
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The Unit Vectors i,
j, k
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Example 1
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Example 2
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Example 3
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Example 4
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Example 4
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Example 5
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Example 6
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Example 7
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Example 8
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Example 9
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Example 10
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Example 11
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problem
واﺟب
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واﺟب
problem
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lecture 3
Vector differentiation
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lecture 4
integration, Gradient,
Divergence and Curl
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lecture 4
Physical significance of
Gradient, Divergence
and Curl.
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lecture 5
curvilinear coordinate
systems.
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Curvilinear coordinates are often
used to define the location or
distribution of physical quantities
which may be, for example, scalars,
vectors, or tensors. Mathematical
expressions involving these
quantities in vector calculus and
tensor analysis (such as the gradient,
divergence, curl, and Laplacian)
Spherical coordinates are
the most common
curvilinear coordinate
systems and are used in
Earth sciences,
cartography, quantum
mechanics, relativity, and
engineering.
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Example
Convert the point (-2,-2,1)from
Cartesian coordinates to
Solution
(a) cylindrical and
(b) spherical coordinates.
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Example:
Find a parametrization of the right
circular cone z2 = x2 +y2 by pulling
back into cylindrical coordinates.
Solution:
Transforming x and y into polar
coordinates yields z2 = r2; z =r
Letting z = r in the cylindrical
coordinate transformation yields
which is a parametrization of
the right circular cone.
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Spherical Coordinates
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Solution
Example
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Example
Solution
IF voltage source was
Equal
find the current in both State
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lecture 6
Gauss and Stoke’s theorem
and their applications
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أﺣﺎدي وﺛﻨﺎﺋﻲ اﻟﻘﻄﺐ
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