Vector Calculus | Engineering Mathematics - Civil Engineering (CE) PDF Download

 Page 1


Applied Engineering Analysis
- slides for class teaching*
Chapter 3
Vectors and Vector Calculus
Chapter Learning Objectives
• To refresh the distinction between scalar and vector quantities in engineering analysis
• To learn the vector calculus and its applications in engineering analysis
• Expressions of vectors and vector functions
• Refresh vector algebra
• Dot and cross products of vectors and their physical meanings
• To learn vector calculus with derivatives, gradient, divergence and curl
• Application of vector calculus in engineering analysis
• Application of vector calculus in rigid body dynamics in rectilinear 
and plane curvilinear motion along paths and in both rectangular
and cylindrical polar coordinate system
Page 2


Applied Engineering Analysis
- slides for class teaching*
Chapter 3
Vectors and Vector Calculus
Chapter Learning Objectives
• To refresh the distinction between scalar and vector quantities in engineering analysis
• To learn the vector calculus and its applications in engineering analysis
• Expressions of vectors and vector functions
• Refresh vector algebra
• Dot and cross products of vectors and their physical meanings
• To learn vector calculus with derivatives, gradient, divergence and curl
• Application of vector calculus in engineering analysis
• Application of vector calculus in rigid body dynamics in rectilinear 
and plane curvilinear motion along paths and in both rectangular
and cylindrical polar coordinate system
Scalar and Vector Quantities
Scalar Quantities:  Physical quantities that have their values determined by the values of the variables 
that define these quantities. For example, in a beam that carries creatures of different 
weight with the forces exerted on the beam determined by the location x only, at 
which the particular creature stands.
X=0
X
5
W(x
5
)
X
W(x)
Vector Quantities: There are physical quantities in engineering analysis, that has their values determined by
NOT only the value of the variables that are associate with the quantities, but also
by the directions that these quantities orient. 
Example of vector quantifies include the
velocities of automobile travelin in winding
street called Lombard Drive in City of San
Francisco the drivers adjusting the velocity
of his(her) automobile according to the 
location of the street with its curvature, but 
also the direction of the automobile that it 
travels on that street.
Page 3


Applied Engineering Analysis
- slides for class teaching*
Chapter 3
Vectors and Vector Calculus
Chapter Learning Objectives
• To refresh the distinction between scalar and vector quantities in engineering analysis
• To learn the vector calculus and its applications in engineering analysis
• Expressions of vectors and vector functions
• Refresh vector algebra
• Dot and cross products of vectors and their physical meanings
• To learn vector calculus with derivatives, gradient, divergence and curl
• Application of vector calculus in engineering analysis
• Application of vector calculus in rigid body dynamics in rectilinear 
and plane curvilinear motion along paths and in both rectangular
and cylindrical polar coordinate system
Scalar and Vector Quantities
Scalar Quantities:  Physical quantities that have their values determined by the values of the variables 
that define these quantities. For example, in a beam that carries creatures of different 
weight with the forces exerted on the beam determined by the location x only, at 
which the particular creature stands.
X=0
X
5
W(x
5
)
X
W(x)
Vector Quantities: There are physical quantities in engineering analysis, that has their values determined by
NOT only the value of the variables that are associate with the quantities, but also
by the directions that these quantities orient. 
Example of vector quantifies include the
velocities of automobile travelin in winding
street called Lombard Drive in City of San
Francisco the drivers adjusting the velocity
of his(her) automobile according to the 
location of the street with its curvature, but 
also the direction of the automobile that it 
travels on that street.
Graphic and mathematical Representation of Vector Quantities
Graphic Representation of a Vector A:
A –ve sign attached to vector A means the
Vector orients in OPPOSITE direction 
A vector A is represented by 
magnitude A in the direction 
shown by arrow head:
Mathematically it is expressed (in a rectangular coordinates (x,y) as:
With the magnitude expressed by
the length of A:
With the magnitude expressed by the length of A:
and the direction by ?:
Vector quantities can be DECOMPOSED into components as illustrated
2
y
2
x
2
2
A A ? ? ? ?
y x
A A A
x
y
A
A
tan ? ?
With MAGNITUDE:
and DIRECTION:
Vector are usually expressed in BOLDFACED letters, e.g. A for vector A
Page 4


Applied Engineering Analysis
- slides for class teaching*
Chapter 3
Vectors and Vector Calculus
Chapter Learning Objectives
• To refresh the distinction between scalar and vector quantities in engineering analysis
• To learn the vector calculus and its applications in engineering analysis
• Expressions of vectors and vector functions
• Refresh vector algebra
• Dot and cross products of vectors and their physical meanings
• To learn vector calculus with derivatives, gradient, divergence and curl
• Application of vector calculus in engineering analysis
• Application of vector calculus in rigid body dynamics in rectilinear 
and plane curvilinear motion along paths and in both rectangular
and cylindrical polar coordinate system
Scalar and Vector Quantities
Scalar Quantities:  Physical quantities that have their values determined by the values of the variables 
that define these quantities. For example, in a beam that carries creatures of different 
weight with the forces exerted on the beam determined by the location x only, at 
which the particular creature stands.
X=0
X
5
W(x
5
)
X
W(x)
Vector Quantities: There are physical quantities in engineering analysis, that has their values determined by
NOT only the value of the variables that are associate with the quantities, but also
by the directions that these quantities orient. 
Example of vector quantifies include the
velocities of automobile travelin in winding
street called Lombard Drive in City of San
Francisco the drivers adjusting the velocity
of his(her) automobile according to the 
location of the street with its curvature, but 
also the direction of the automobile that it 
travels on that street.
Graphic and mathematical Representation of Vector Quantities
Graphic Representation of a Vector A:
A –ve sign attached to vector A means the
Vector orients in OPPOSITE direction 
A vector A is represented by 
magnitude A in the direction 
shown by arrow head:
Mathematically it is expressed (in a rectangular coordinates (x,y) as:
With the magnitude expressed by
the length of A:
With the magnitude expressed by the length of A:
and the direction by ?:
Vector quantities can be DECOMPOSED into components as illustrated
2
y
2
x
2
2
A A ? ? ? ?
y x
A A A
x
y
A
A
tan ? ?
With MAGNITUDE:
and DIRECTION:
Vector are usually expressed in BOLDFACED letters, e.g. A for vector A
3.2 Vectors expressed in terms of Unit Vectors in Rectangular coordinate Systems
- A simple and convenient way to express vector quantities 
Let:   i = unit vector along the x-axis
j = unit vector along the y-axis
k = unit vector along the z-axis 
in a rectangular coordinate system (x,y,z), or
a cylindrical polar coordinate system (r, ?,z).
All unit vectors i,  j and k have a magnitudes of 1.0 (i.e. unit)
Then the position vector A (with it “root” coincides with th origin of the coordinate system)
expressed in the following form:
A = xi + yj + zk
where x = magnitude of the component of Vector A in the x-coordinate
y = magnitude of the component of Vector A in the y-coordinate
z = magnitude of the component of Vector A in the z-coordinate
? ?
2 2 2 2
2
2 2
z y x z y x A ? ? ? ? ? ? ? A
We may thus evaluate the magnitude of the vector A to be the sum of the magnitudes of all its components as: 
Page 5


Applied Engineering Analysis
- slides for class teaching*
Chapter 3
Vectors and Vector Calculus
Chapter Learning Objectives
• To refresh the distinction between scalar and vector quantities in engineering analysis
• To learn the vector calculus and its applications in engineering analysis
• Expressions of vectors and vector functions
• Refresh vector algebra
• Dot and cross products of vectors and their physical meanings
• To learn vector calculus with derivatives, gradient, divergence and curl
• Application of vector calculus in engineering analysis
• Application of vector calculus in rigid body dynamics in rectilinear 
and plane curvilinear motion along paths and in both rectangular
and cylindrical polar coordinate system
Scalar and Vector Quantities
Scalar Quantities:  Physical quantities that have their values determined by the values of the variables 
that define these quantities. For example, in a beam that carries creatures of different 
weight with the forces exerted on the beam determined by the location x only, at 
which the particular creature stands.
X=0
X
5
W(x
5
)
X
W(x)
Vector Quantities: There are physical quantities in engineering analysis, that has their values determined by
NOT only the value of the variables that are associate with the quantities, but also
by the directions that these quantities orient. 
Example of vector quantifies include the
velocities of automobile travelin in winding
street called Lombard Drive in City of San
Francisco the drivers adjusting the velocity
of his(her) automobile according to the 
location of the street with its curvature, but 
also the direction of the automobile that it 
travels on that street.
Graphic and mathematical Representation of Vector Quantities
Graphic Representation of a Vector A:
A –ve sign attached to vector A means the
Vector orients in OPPOSITE direction 
A vector A is represented by 
magnitude A in the direction 
shown by arrow head:
Mathematically it is expressed (in a rectangular coordinates (x,y) as:
With the magnitude expressed by
the length of A:
With the magnitude expressed by the length of A:
and the direction by ?:
Vector quantities can be DECOMPOSED into components as illustrated
2
y
2
x
2
2
A A ? ? ? ?
y x
A A A
x
y
A
A
tan ? ?
With MAGNITUDE:
and DIRECTION:
Vector are usually expressed in BOLDFACED letters, e.g. A for vector A
3.2 Vectors expressed in terms of Unit Vectors in Rectangular coordinate Systems
- A simple and convenient way to express vector quantities 
Let:   i = unit vector along the x-axis
j = unit vector along the y-axis
k = unit vector along the z-axis 
in a rectangular coordinate system (x,y,z), or
a cylindrical polar coordinate system (r, ?,z).
All unit vectors i,  j and k have a magnitudes of 1.0 (i.e. unit)
Then the position vector A (with it “root” coincides with th origin of the coordinate system)
expressed in the following form:
A = xi + yj + zk
where x = magnitude of the component of Vector A in the x-coordinate
y = magnitude of the component of Vector A in the y-coordinate
z = magnitude of the component of Vector A in the z-coordinate
? ?
2 2 2 2
2
2 2
z y x z y x A ? ? ? ? ? ? ? A
We may thus evaluate the magnitude of the vector A to be the sum of the magnitudes of all its components as: 
Examples of using unit vectors in engineering analysis
Example 3.1: A vector A in Figure 3.2(b) has its two components along the x- and y-axis with respective 
magnitudes of 6 units and 4 units. Find the magnitude and direction of the vector A.
Solution: Let us first illustrate the vector A in the x-y plane:
x
y
A
x=6
y=4
0
The vector A may be expressed in terms of unit vectors I and j as:
A = 6i + 4J
And the magnitude of vector A is:
P
and the angle ? is obtained by:
?
A Vector in 3-D Space in a Rectangular coordinate System:
0
A
y
= y
X 
y
z
P(x,y,z)
A
z
=z
The vector A may be expressed in terms of unit vectors i, j and k  as:
A = xi + yj + zk
where x = magnitude of the component of Vector A in the x-coordinate
y = magnitude of the component of Vector A in the y-coordinate
z = magnitude of the component of Vector A in the z-coordinate
The magnitude of vector A is:
The direction of the vector is determined by: