1. Review of Vector Calculus
Lecture 1
Review of Vector Calculus

2. In this lecture: Vectors Three-Dimensional Coordinate Systems
Dot Product
Cross Product
Line Integral
Surface Integral
Volume Integral

3. Vectors

4. Scalars are quantities having only a magnitude.
Length, mass, temperature etc.
VECTORS
The term vector is used by scientists to indicate a quantity (such as displacement or velocity or force) that has both magnitude and direction.
A vector is often represented by an arrow or a directed line segment.
We denote a vector by printing a letter in boldface (v) or by putting an arrow above the letter (v).

5. Length represents magnitude. Arrow head represents direction.
VECTOR REPRESENTATION
Arrow head represents
direction.
Length represents
magnitude.
Figure 1. Vector representation

6. Force Vector Line of force (or pull). Tail represents point of force
EXAMPLE: FORCE VECTOR
Tail represents
point of force
application.
Line of force
(or pull).
Force Vector
Another way to represent vector quantities is graphically, with an arrow. <ENTER> By using arrows to represent vector quantities, we can draw diagrams of movement situations to help us systematically analyze the movement. This is especially useful for analyzing forces. <ENTER> The arrow must depict both characteristics of the vector quantity – the magnitude and the direction. <ENTER> The magnitude of the vector quantity is depicted by the length of the arrow. To be accurate in depicting magnitude, you would have to set up a conversion scale, much like what is used in road maps. For example, if you state that a 1 inch vector represents 50 N of force, then you would draw a 2 inch vector to represent a 100 N force. While we will not formally scale vectors this semester, you should be comfortable with this concept so that you can estimate forces from vectors that are given to you, or so that you can make general representations of force magnitudes when you are drawing and comparing two forces. <ENTER> The second characteristic – the direction – is represented by the arrow head of the vector. Obviously, this will be placed on the end of the vector that indicates the direction associated with the quantity. <ENTER> Because we will be dealing with force vectors quite a bit this semester, I want to introduce two other characteristics of forces that should also be depicted on vectors. <ENTER> First, is the point of force application. When drawing a vector to represent a force, the tail of the vector should be placed on the point of force application. <ENTER> Second, a dashed line that runs towards infinity along the line of the vector should also be drawn to indicate the line of force for the force being analyzed. This is also sometimes called the line of pull, if the force being analyzed is a pulling force, like a muscle force. The importance of this characteristic will be explained better when we discuss torque in our next lecture. Now, let’s look at some examples of vector representation. <ENTER>
Figure 2. Force vector 6

7. COORDINATE SYSTEMS

8. CARTESIAN COORDINATE SYSTEM
has three orthogonal axes
axes are labelled as x, y, and z.
O is origin.
Figure 3. Coordinate axes

9. LABELLING AXES Use the right-hand rule when labelling axes.
thumb ≡ z-axis
first finger ≡ x-axis
second finger ≡ y-axis
Figure 4. Right-hand rule

10. COORDINATE PLANES
Figure 5. Coordinate planes

11. COORDINATES OF POINT P Coordinates of point P written as P(x,y,z).
Figure 6. Coordinates of point P.

12. A1, A2 and A3 are called x, y, and z component of vector A
Position vector of a point in space
A point P in Cartesian coordinate system may be expressed as its x,y,z coordinates. The position vector of a point P is the directed distance from the origin O to the point P.
A = A1i + A2j + A3k = (A1, A2, A3) P
A1, A2 and A3 are called x, y, and z component of vector A
i, j, and k are unit vectors pointing in the positive x, y, and z directions

13. is a unit directional vector !
Position vector of a point in space
Magnitude of A:
Direction of A:
is a unit directional vector !
Note:

14. Example 1
Give the position vectors for points A and B, their magnitudes and directions(c) The position vector for 
Solution
The position vector of a point is nothing more than a vector with the point's coordinates as its components. Therefore,

15. Solution
Magnitude of A:
Direction of A:

16. Solution
Magnitude of B:
Direction of B:

17. Displacement Vector
A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P. A 'displacement vector' represents the length and direction of that imaginary straight path.

18. A displacement may be also described as a 'relative position': the final position of a point (Q) relative to its initial position (P), and a displacement vector can be mathematically defined as the difference between the final and initial position vectors:
PQ = OQ - OP P Q x y OP OQ PQ

19. Example
The position vector of P is (4,3) and the position vector of Q is (-2,5). Find the displacement vector PQ. x y
P(4,-3)
Q(-2,5)

20. Solution
PQ = OQ - OP x y
P(4,-3)
Q(-2,5)

21. CYLINDRICAL COORDINATE SYSTEM
Any point in space is considered to be at the intersection of three mutually perpendicular surfaces:
A circular cylinder (r = constant)
A vertical plane (  = constant)
A horizontal plane ( z = constant) r

22. Cylindrical Coordinate System
Any point in space is represented by three coordinates P(r,,z)
r denotes the radius of an imaginary cylinder passing through P, or the radial distance from z axis to the point P
 Denotes azimuthal angle, measured from x axis to a vertical intersecting plane passing through P
z denotes distance from xy-plane to a horizontal intersecting plane passing through P. It is the same as in rectangular coordinate system.
z = constant r r
r = constant r r
=constant

23. A vector in cylindrical coordinate system may be specified using three mutually perpendicular unit vectors
form a right-handed system because an RH screw when rotated from r to  moves towards z.
These unit vector specify directions along r, , and z axes.
Using these unit vectors any vector A may be expressed as
The magnitude of the vectors is given by
CYLINDRICAL COORDINATE SYSTEMS

24. Visualisation of point P(3,45o,8) in the cylindrical coordinate system

25. SPHERICAL COORDINATE SYSTEM
Any point in space is considered to be at the intersection of three mutually perpendicular surfaces:
A sphere of radius r from the origin (r = constant)
A cone centred around the z axis (  = constant)
A vertical (  = constant)
Any point in spherical coordinate system is considered to be at the intersection of the above three planes.

26. SPHERICAL COORDINATE SYSTEM
A vector in spherical coordinate system may be specified using three mutually perpendicular unit vectors
form a right-handed system because an RH screw when rotated from r to  moves towards z.
These unit vector specify directions along r, , and  axes.
Using these unit vectors any vector A may be expressed as
The magnitude of the vectors is given by

28. DEFINITION OF SCALAR MULTIPLICATION
If c is a scalar and v is a vector, then the scalar multiple cv is the vector whose length is c times the length of v and whose direction is the same as v if c > 0 and is opposite to v if c < 0. If c = 0 or v = 0, then cv = 0.
Notice that two nonzero vectors are parallel if they are scalar multiples of one another. In particular, the vector –v=(-1)v has the same length as v but points in the opposite direction. We call it the negative of v.

29. DEFINITION OF SCALAR MULTIPLICATION
If c is a scalar and v is a vector, then the scalar multiple cv is the vector whose length is c times the length of v and whose direction is the same as v if c > 0 and is opposite to v if c < 0. If c = 0 or v = 0, then cv = 0.
Notice that two nonzero vectors are parallel if they are scalar multiples of one another. In particular, the vector –v=(-1)v has the same length as v but points in the opposite direction. We call it the negative of v.

30. EXAMPLE OF SCALAR MULTIPLICATION
From the similar triangles below we see that the components of ca are ca1 and ca2. So to multiply a vector by a scalar we multiply each component by that scalar.

31. EQUIVALENT VECTORS
Displacement vector v , shown in Figure 1, has initial point A (the tail) and terminal point B (the tip) and we indicate this by writing v=AB. Notice that the vector u=CD has the same length and the same direction as v even though it is in a different position. We say that u and v are equivalent (or equal) and we write u=v.

32. DEFINITION OF VECTOR ADDITION
If u and v are vectors positioned so the initial point of v is at the terminal point of u, then the sum u + v is the vector from the initial point of u to the terminal point of v.
To add algebraic vectors we add their components.

33. EXAMPLE If a=<a1,a2> and b =<b1,b2>, then the sum is
a + b=<a1+b1, a2+b2>

34. PARALLELOGRAM LAW

35. DIFFERENCE OF TWO VECTORS
By the difference u - v of two vectors we mean
u - v= u + (-v)
Drawing u – v:
To subtract vectors we subtract components.

36. EXAMPLE If a=<a1,a2> and b=<b1,b2>, then
Similarly, for three-dimensional vectors,
a – b = <a1 – b1, a2 – b2>
a – b = <a1, a2, a3> - <b1, b2, b3>
= <a1- b1, a2 - b2, a3 - b3>

37. PROPERTIES OF VECTORS
If a, b, and c are vectors and d is a scalar, then
a + b = b + a (Commutative law)
a + (b + c) = ( a + b ) + c (Associative law)
a + 0 = a
a + (-a) = 0
d(a + b) = da + db
a = a

38. GRAPHICAL PROOF

39. DOT PRODUCT Definition
If a = <a1,a2,a3> and b = <b1,b2,b3> , then the dot product of a and b is a scalar c given by
c = a.b = a1b1 + a2b2 + a3b3
Another name for dot product is scalar product.
Example
If a = (1, -3, 2) and b = (4, 5, -8), find the dot product of a and b.
Solution
c = a.b = a1b1 + a2b2 + a3b3
= (1)x(1) + (-3)x(5) + (2)x(-8)
= - 27

40. PROPERTIES OF THE DOT PRODUCT
If a, b, and c are 3-D vectors, then
a.a = |a|2
a.b = b.a
a.(b + c) = a.b + a.c
(c.a).b = c.(a.b) = a.(c.b)
0.a = 0

41. Geometric Interpretation of Dot Product
If θ is the angle between the nonzero vectors a and b, then
a.b = |a||b|cosθ or θ a b
cos θ = a.b / |a||b|
Note
Two non-zero vectors a and b are orthogonal if and only if
a‧b = 0

42. CROSS PRODUCT OF TWO VECTORS
If a = a1i + a2j + a3k and b = b1i+ b2j + b3k, then the cross product of a and b is written as a x b.
The cross product a x b is a vector c such that θ b a c
Note
The vector c is orthogonal to both a and b.

43. Example:
Let A = (1, -3, 2) and B = (4, 5, -8), then

44. GEOMETRIC INTEPRETATION
If θ is the angle between the nonzero vectors a and b, then the cross product of a and b is a vector c whose magnitude is given by the expression
c = a x b = |a||b|sinθ
and whose direction is given by the right hand rule. c b a
Note
Two nonzero vectors a and b are parallel if and only if a x b = 0

48. i x j = k ; j x k = I k x i = j ; j x i = -k k x j = -i ; i x k = -j
CROSS PRODUCT OF BASIS VECTORS
i x j = k ; j x k = I
k x i = j ; j x i = -k
k x j = -i ; i x k = -j
i x j j x i

49. If a and b are vectors and c is a scalar, then
THEOREM
If a and b are vectors and c is a scalar, then
a x b = -b x a
(ca) x b = c(a x b) = a x (cb)

52. The basics

54. VECTOR AND SCALAR FUNCTIONS
A vector valued function A(t) is a rule that associates with each real number t a vector A(t).
A(t) = A1(t)i + A2(t)j + A3(t)k
For example, f(t) = t3 – 2t + 4 is a scalar function of a single variable t, while A(t) = cos ti + sin tj + tk is a vector function of t. 54

55. A′(t) = A1′(t)i + A2′(t)j + A3′(t)k
VECTOR DIFFERENTIATION
A vector function A(t) is differentiable at a point t if
exists, and A′(t) is called the derivative of A(t), written as
A′(t) = A1′(t)i + A2′(t)j + A3′(t)k
Calculate the derivative of each component!

56. Example:
Let A(t) = cos ti + sin tj + tk. Find the derivative of A(t).
Solution:
A′(t) = -sin ti + cos tj + k

57. RULES OF VECTOR DIFFERENTIATION
if A = constant. 57

58. VECTOR INTEGRATION
Let A(t) = A1(t)i + A2(t)j + A3(t)k and suppose that the component functions A1(t), A2(t) and A3(t) are integrable. Then the indefinite integral of A(t) is defined by
Calculate the integral of each component!
If A1(t), A2(t) and A3(t) are integrable over the interval [t1, t2], then the definite integral of A(t) is defined by

59. Example
Let A(t) = cos ti + sin tj + tk. Find
Solution:

60. SCALAR FIELD
If every point in a region of space is assigned a scalar value obtained from a scalar function f(x, y, z), then a scalar field f(x, y, z) is defined in the region, such as the pressure in atmosphere and mass density within the earth, etc.

61. Partial Derivatives
Mixed second partials

62. Example
Let f = x2 + 2y2. Calculate and
Solution:

63. GRADIENT Del operator Gradient
Gradient characterizes maximum increase. If at a point P the gradient of f is not the zero vector, it represents the direction of maximum space rate of increase in f at P.

64. Example
Given potential function V = x2y + xy2 + xz2, (a) find the gradient of V, and (b) evaluate it at (1, -1, 3).
Solution:
(a)
(b)
Direction of maximum increase

65. Electric field: E = E(x, y, z),
VECTOR FIELD
Electric field: E = E(x, y, z),
Magnetic field : H = H(x, y, z)
If every point in a region of space is assigned a vector value obtained from a vector function A(x, y, z), then a vector field A(x, y, z) is defined in the region.

66. DIVERGENCE OF A VECTOR FIELD
The divergence of a vector field A at a point is defined as the net outward flux of A per unit volume as the volume about the point tends to zero:
It indicates the presence of a source (or sink)!  term the source as flow source. And div A is a measure of the strength of the flow source.
Representing field variations graphically by directed field lines - flux lines

67. div A = 3z + 2x – 2yz DIVERGENCE OF A VECTOR FIELD
In rectangular coordinate, the divergence of A can be calculated as
For instance, if A = 3xzi + 2xyj – yz2k, then
div A = 3z + 2x – 2yz
At (1, 2, 2), div A = 0; at (1, 1, 2), div A = 4, there is a source; at (1, 3, 1), div A = -1, there is a sink.

68. CURL OF A VECTOR FIELD
The curl of a vector field A is a vector whose magnitude is the maximum net circulation of A per unit area as the area tends to zero and whose direction is the normal direction of the area.
It is an indication of a vortex source, which causes a circulation of a vector field around it.
Water whirling down a sink drain is an example of a vortex sink causing a circulation of fluid velocity.
If A is electric field intensity, then the circulation will be an electromotive force around the closed path.
Water vortex

69. CURL OF A VECTOR FIELD
In rectangular coordinate, curl A can be calculated as

70. Example:
If A = yzi + 3zxj + zk, then

71. circulation of E around L
LINE INTEGRAL OF VECTOR FUNCTIONS z D C dl dl y E x
; dl = dxi +dyj +dzk
For a closed loop, i.e. ABCA,
circulation of E around L

72. Example
For F = yi – xj, calculate the circulation of F along the two paths as shown below.
Solution:
dl = dxi +dyj +dzk
Along path C2

73. SURFACE INTEGRAL E
Surface integral or the flux of E across the surface S is
is the outward unit vector normal to the surface.
For closed surface,
net outward flux of E.

74. Example
If F = xi + yj + (z2 – 1)k, calculate the flux of F across the surface shown in the figure.
Solution:

75. VOLUME INTEGRAL
Evaluation : choose a suitable integration order and then find out the suitable lower and upper limits for x, y and z respectively.

76. Example: Let F = 2xzi – xj + y2k. Evaluate
where V is the region bounded by the surface x = 0, x = 2, y = 0, y = 6, z = 0, z = 4.
Solution:

77. where v = volume charge density (C/m3)
In electromagnetic,
= total charge within the volume
where v = volume charge density (C/m3)