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3. VECTOR ANALYSISRuzelita Ngadiran
OverviewBasic Laws of Vector AlgebraDot Product and Cross ProductOrthogonal Coordinate Systems: Cartesian, Cylindrical and Spherical Coordinate SystemsTransformations between Coordinate SystemsGradient of a Scalar FieldDivergence of a Vector FieldDivergence TheoremCurl of a Vector FieldStokess TheoremLaplacian Operator
This chapter cover CO1Ability to describe different coordinate system and their interrelation.
ScalarA scalar is a quantity that has only magnitudeE.g. of Scalars:Time, mass, distance, temperature, electrical potential etc
VectorA vector is a quantity that has both magnitude and direction.E.g. of Vectors:Velocity, force, displacement, electric field intensity etc.
Basic Laws of Vector AlgebraCartesian coordinate systems
Vector in Cartesian CoordinatesA vector in Cartesian Coordinates maybe represented as
OR
Vector in Cartesian CoordinatesVector A has magnitude A = |A| to the direction of propagation.Vector A shown may be represented as
The vector A has three component vectors, which are Ax, Ay and Az.
Laws of Vector Algebra stopmagnitudeUnit vectormagnitudeUnit vector
Example 1 : Unit VectorSpecify the unit vector extending from the origin towards the point
Solution :Construct the vector extending from origin to point G
Find the magnitude of
Solution :So, unit vector is
Properties of Vector OperationsEquality of Two Vectors
Vector AlgebraFor addition and subtraction of A and B,Hence,Commutative property
Example 2 : If
Find:
Solution to Example 2(a)The component of along is (b)
ContHence, the magnitude of is:(c) Let
ContSo, the unit vector along is:
Position & Distance VectorsPosition Vector: From origin to point PDistance Vector: Between two points
Position and distance Vector
Example 3Point P and Q are located at and . Calculate: The position vector P The distance vector from P to Q The distance between P and Q A vector parallel to with magnitude of 10
Solution to Example 3*(a)(b)
(c)
Since is a distance vector, the distance between P and Q is the magnitude of this distance vector.UNIVERSITI MALAYSIA PERLIS The position vector P The distance vector from P to Q The distance between P and Q A vector parallel to with magnitude of 10
Solution to Example 3*Distance, d (d)
Let the required vector be thenWhere is the magnitude of UNIVERSITI MALAYSIA PERLIS
Solution to Example 3*Since is parallel to , it must have the same unit vector as orSo, UNIVERSITI MALAYSIA PERLIS
Multiplication of Vectors* When two vectors and are multiplied, the result is either a scalar or vector, depending on how they are multiplied. Two types of multiplication: Scalar (or dot) product Vector (or cross) productUNIVERSITI MALAYSIA PERLIS
Scalar or Dot Product*The dot product of two vectors, and is defined as the product of the magnitude of , the magnitude of and the cosine of the smaller angle between them.UNIVERSITI MALAYSIA PERLIS
Dot Product in Cartesian* The dot product of two vectors of Cartesian coordinate below yields the sum of nine scalar terms, each involving the dot product of two unit vectors.UNIVERSITI MALAYSIA PERLIS
Dot Product in Cartesian* Since the angle between two unit vectors of the Cartesian coordinate system is , we then have: And thus, only three terms remain, giving finally:UNIVERSITI MALAYSIA PERLIS
Dot Product in CartesianThe two vectors, and are said to be perpendicular or orthogonal (90) with each other if;
*UNIVERSITI MALAYSIA PERLIS
Laws of Dot ProductDot product obeys the following:
Commutative Law
Distributive Law
*UNIVERSITI MALAYSIA PERLIS
Properties of dot product*
Properties of dot product of unit vectors:UNIVERSITI MALAYSIA PERLIS
Vector Multiplication: Scalar Product or Dot Product
Hence:
Vector or Cross Product* The cross product of two vectors, and is a vector, which is equal to the product of the magnitudes of and and the sine of smaller angle between themUNIVERSITI MALAYSIA PERLIS
Vector or Cross Product*Direction of is perpendicular (90) to the plane containing A and B
Vector or Cross ProductIt is also along one of the two possible perpendiculars which is in direction of advance of right hand screw.
*UNIVERSITI MALAYSIA PERLIS
Cross product in Cartesian* The cross product of two vectors of Cartesian coordinate:
yields the sum of nine simpler cross products, each involving two unit vectors.UNIVERSITI MALAYSIA PERLIS
Cross product in Cartesian* By using the properties of cross product, it givesand be written in more easily remembered form:UNIVERSITI MALAYSIA PERLIS
Laws of Vector Product* Cross product obeys the following: It is not commutative It is not associative It is distributiveUNIVERSITI MALAYSIA PERLIS
Properties of Vector Product*Properties of cross product of unit vectors:Or by using cyclic permutation:UNIVERSITI MALAYSIA PERLIS
Vector Multiplication: Vector Product or Cross Product
Example 4:Dot & Cross Product*Determine the dot product and cross product of the following vectors:UNIVERSITI MALAYSIA PERLIS
Solution to Example 4*The dot product is:UNIVERSITI MALAYSIA PERLIS
Solution to Example 4*The cross product is:UNIVERSITI MALAYSIA PERLIS
Scalar & Vector Triple Product A scalar triple product is
A vector triple product is
known as the bac-cab rule.
*UNIVERSITI MALAYSIA PERLIS
Triple ProductsScalar Triple ProductVector Triple Product
Example 5
Given , and .
Find (AB)C and compare it with A(BC). *UNIVERSITI MALAYSIA PERLIS
Solution to Example 5
A similar procedure gives *UNIVERSITI MALAYSIA PERLIS
ContHence :
Example From Book Scalar/ dot product
Solution
Solution
Cont
Coordinate SystemsCartesian coordinates
Circular Cylindrical coordinates
Spherical coordinates
*UNIVERSITI MALAYSIA PERLIS
Cartesian coordinatesConsists of three mutually orthogonal axes and a point in space is denoted as
*UNIVERSITI MALAYSIA PERLIS
Cartesian CoordinatesUnit vector of in the direction of increasing coordinate value.
*UNIVERSITI MALAYSIA PERLIS
Cartesian CoordinatesDifferential in Length
*UNIVERSITI MALAYSIA PERLIS
Cartesian CoordinatesDifferential Surface*UNIVERSITI MALAYSIA PERLIS
Cartesian CoordinatesDifferential Surface
*UNIVERSITI MALAYSIA PERLIS
Cartesian CoordinatesDifferential Volume*UNIVERSITI MALAYSIA PERLIS
Cartesian Coordinate SystemDifferential length vectorDifferential area vectors
Circular Cylindrical Coordinates*UNIVERSITI MALAYSIA PERLIS
Circular Cylindrical CoordinatesForm by three surfaces or planes: Plane of z (constant value of z) Cylinder centered on the z axis with a radius of . Some books use the notation . Plane perpendicular to x-y plane and rotate about the z axis by angle of Unit vector of in the direction of increasing coordinate value.
*UNIVERSITI MALAYSIA PERLIS
*Differential in LengthCircular Cylindrical CoordinatesUNIVERSITI MALAYSIA PERLIS
Circular Cylindrical CoordinatesIncrement in length for direction is:
is not increment in length!
*UNIVERSITI MALAYSIA PERLIS
Circular Cylindrical CoordinatesDifferential Surface*UNIVERSITI MALAYSIA PERLIS
Circular Cylindrical CoordinatesDifferential volume*UNIVERSITI MALAYSIA PERLIS
Calculus Basic
Cylindrical Coordinate System
Cylindrical Coordinate System
Example 6 A cylinder with radius of and length of Determine:
(i) The volume enclosed.
(ii) The surface area of that volume.
*UNIVERSITI MALAYSIA PERLIS
FORMULADifferential volume*UNIVERSITI MALAYSIA PERLIS
Solution to Example 6 (i) For volume enclosed, we integrate;
*UNIVERSITI MALAYSIA PERLIS
*Differential SurfaceFORMULAUNIVERSITI MALAYSIA PERLIS
Solution to Example 6 (ii) For surface area, we add the area of each surfaces;
*UNIVERSITI MALAYSIA PERLIS
Example 7The surfaces define a closed surface. Find:
The enclosed volume. The total area of the enclosing surface.*UNIVERSITI MALAYSIA PERLIS
Solution to Example 7(a) The enclosed volume;
*Must convert into radiansUNIVERSITI MALAYSIA PERLIS
Solution to Example 7(b) The total area of the enclosed surface:*UNIVERSITI MALAYSIA PERLIS
From Book
From Book
A circular cylinder of radius r = 5 cm is concentric with the z-axis and extends between z = 0 cm and z = 3 cm. find the cylinders volume.= 471.2 cm^3EXERCISE 1 2
Spherical Coordinates*UNIVERSITI MALAYSIA PERLIS
Spherical CoordinatesPoint P in spherical coordinate,
distance from origin. Some books use the notation
angle between the z axis and the line from origin to point P
angle between x axis and projection in z=0 plane
*UNIVERSITI MALAYSIA PERLIS
Spherical CoordinatesUnit vector of in the direction of increasing coordinate value.
*UNIVERSITI MALAYSIA PERLIS
Spherical CoordinatesDifferential in length
*UNIVERSITI MALAYSIA PERLIS
Spherical CoordinatesDifferential Surface *UNIVERSITI MALAYSIA PERLIS
Spherical CoordinatesDifferential Surface*UNIVERSITI MALAYSIA PERLIS
Spherical CoordinatesDifferential Volume*UNIVERSITI MALAYSIA PERLIS
Spherical CoordinatesHowever, the increment of length is different from the differential increment previously, where:
distance between two radius distance between two angles distance between two radial planes at angles*UNIVERSITI MALAYSIA PERLIS
Spherical Coordinate System
Example 8a*A sphere of radius 2 cm contains a volume chargedensity v given by;
Find the total charge Q contained in the sphere.UNIVERSITI MALAYSIA PERLIS
Solution: Example 8a*UNIVERSITI MALAYSIA PERLIS
Example 8bThe spherical strip is a section of a sphere of radius 3 cm. Find the area of the strip.
*UNIVERSITI MALAYSIA PERLIS
Solution : Example 8bUse the elemental area with constant R, that is . Solution:*UNIVERSITI MALAYSIA PERLIS
ExerciseAnswer
Coordinate Transformations: CoordinatesTo solve a problem, we select the coordinate system that best fits its geometrySometimes we need to transform between coordinate systems
Coordinate Transformations: Unit Vectors
Cartesian to Cylindrical Transformations*Relationships between (x, y, z) and (r, , z) are shown.UNIVERSITI MALAYSIA PERLIS
*Cartesian to Spherical Transformations
Relationships between (x, y, z) and (r, , ) are shown in the diagram.UNIVERSITI MALAYSIA PERLIS
*Cartesian to Spherical Transformations
Relationships between (x, y, z) and (r, , ) are shown.UNIVERSITI MALAYSIA PERLIS
Example 9*Express vector in spherical coordinates.
Using the transformation relation,
Using the expressions for x, y, and z,Solution
Example 9: contd*Similarly, substituting the expression for x, y, z for;
we get:
Hence,UNIVERSITI MALAYSIA PERLIS
Ex: Cartesian to Cylindrical in degree
Distance Between 2 Points
TransformationsDistance d between two points is
Converting to cylindrical equivalents
Converting to spherical equivalents
*UNIVERSITI MALAYSIA PERLIS
Gradient of a scalar field*Suppose is the temperature at ,and is the temperature at as shown. UNIVERSITI MALAYSIA PERLIS
Gradient of a scalar field*The differential distances are the components of the differential distance vector :However, from differential calculus, the differential temperature:UNIVERSITI MALAYSIA PERLIS
Gradient of a scalar field*But,So, previous equation can be rewritten as:UNIVERSITI MALAYSIA PERLIS
Gradient of a scalar field*The vector inside square brackets defines the change of temperature corresponding to a vector change in position .This vector is called Gradient of Scalar T.For Cartesian coordinate, grad T:
The symbol is called the del or gradient operator.
UNIVERSITI MALAYSIA PERLIS
Gradient operator in cylindrical and spherical coordinatesGradient operator in cylindrical coordinates:
Gradient operator in spherical coordinates:
*UNIVERSITI MALAYSIA PERLISAfter this, Go to slide 115
Gradient of A Scalar Field
Gradient ( cont.)
Example 10Find the gradient of these scalars:*(a)
(b)
(c)UNIVERSITI MALAYSIA PERLIS
Solution to Example 10*(a) Use gradient for Cartesian coordinate:UNIVERSITI MALAYSIA PERLIS
Solution to Example 10*(b) Use gradient for cylindrical coordinate:UNIVERSITI MALAYSIA PERLIS
Solution to Example 10*(c) Use gradient for Spherical coordinate:UNIVERSITI MALAYSIA PERLIS
Directional derivative tahun 2Gradient operator del, has no physical meaning by itself.Gradient operator needs to be scalar quantity.Directional derivative of T is given by,
*UNIVERSITI MALAYSIA PERLIS
Example 11*Find the directional derivative of
along the direction and evaluate it at (1,1, 2).
UNIVERSITI MALAYSIA PERLIS
Solution to Example 11*
GradT :
We denote L as the given direction,
Unit vector is
andUNIVERSITI MALAYSIA PERLIS
Divergence of a vector fieldIllustration of the divergence of a vector field at point P:
*Positive DivergenceNegative DivergenceZero DivergenceUNIVERSITI MALAYSIA PERLIS
Divergence of a vector fieldThe divergence of A at a given point P is the net outward flux per unit volume:
*UNIVERSITI MALAYSIA PERLIS
Divergence of a vector field*What is ?? Vector field A at closed surface SUNIVERSITI MALAYSIA PERLIS
Divergence of a vector field*Where,And, v is volume enclosed by surface SUNIVERSITI MALAYSIA PERLIS
Divergence of a vector field*For Cartesian coordinate:For Circular cylindrical coordinate:UNIVERSITI MALAYSIA PERLIS
Divergence of a vector field*For Spherical coordinate:UNIVERSITI MALAYSIA PERLIS
Divergence of a vector field Tahun 4Example: A point charge qTotal flux of the electric field E due to q is
*UNIVERSITI MALAYSIA PERLIS
Divergence of a vector field*Net outward flux per unit volume i.e the div of E is
UNIVERSITI MALAYSIA PERLIS
Example 12Find divergence of these vectors:
*(a)
(b)
(c)UNIVERSITI MALAYSIA PERLIS
Solution to Example 12(a) Use divergence for Cartesian coordinate:*UNIVERSITI MALAYSIA PERLIS
Solution to Example 12(b) Use divergence for cylindrical coordinate:
*UNIVERSITI MALAYSIA PERLIS
Solution to Example 12(c) Use divergence for Spherical coordinate:*UNIVERSITI MALAYSIA PERLIS
Divergence of a Vector Field
Divergence TheoremUseful tool for converting integration over a volume to one over the surface enclosing that volume, and vice versa
Curl of a Vector Field
Curl of a vector fieldThe curl of vector A is an axial (rotational) vector whose magnitude is the maximum circulation of A per unit area Curl direction is the normal direction of the area when the area is oriented so as to make the circulation maximum.*UNIVERSITI MALAYSIA PERLIS
Curl of a vector fieldThe circulation of B around closed contour C:*UNIVERSITI MALAYSIA PERLIS
Curl of a vector fieldCurl of a vector field B is defined as:*UNIVERSITI MALAYSIA PERLIS
Curl of a vector fieldCurl is used to measure the uniformity of a fieldUniform field, circulation is zeroNon-uniform field, e.g azimuthal field, circulation is not zero*UNIVERSITI MALAYSIA PERLIS
Curl of a vector fieldUniform field, circulation is zero
*UNIVERSITI MALAYSIA PERLIS
Curl of a vector fieldNon-uniform field, e.g azimuthal field, circulation is not zero
*UNIVERSITI MALAYSIA PERLIS
Vector identities involving curlFor any two vectors A and B:
*UNIVERSITI MALAYSIA PERLIS
Curl in Cartesian coordinatesFor Cartesian coordinates:
*UNIVERSITI MALAYSIA PERLIS
Curl in cylindrical coordinatesFor cylindrical coordinates:
*UNIVERSITI MALAYSIA PERLIS
Curl in spherical coordinatesFor spherical coordinates:*UNIVERSITI MALAYSIA PERLIS
Example 14Find curl of these vectors:
*(a)
(b)
(c)UNIVERSITI MALAYSIA PERLIS
Solution to Example 14(a) Use curl for Cartesian coordinate:
*UNIVERSITI MALAYSIA PERLIS
Solution to Example 14(b) Use curl for cylindrical coordinate
*UNIVERSITI MALAYSIA PERLIS
Solution to Example 14(c) Use curl for Spherical coordinate:
*UNIVERSITI MALAYSIA PERLIS
Solution to Example 14*UNIVERSITI MALAYSIA PERLIS
Solution to Example 14(c) continued
*UNIVERSITI MALAYSIA PERLIS
Stokess Theorem
Stokess TheoremConverts surface integral of the curl of a vector over an open surface S into a line integral of the vector along the contour C bounding the surface S*UNIVERSITI MALAYSIA PERLIS
Example 15A vector field is given by . Verify Stokess theorem for a segment of a cylindrical surface defined by r = 2, /3 /2, 0 z 3 as shown in the diagram on the next slide.
*UNIVERSITI MALAYSIA PERLIS
Example 15*UNIVERSITI MALAYSIA PERLIS
Solution to Example 15Stokess theorem states that:
Left-hand side: First, use curl in cylindrical coordinates
*UNIVERSITI MALAYSIA PERLIS
Solution to Example 15The integral of over the specified surface S with r = 2 is:
*UNIVERSITI MALAYSIA PERLIS
Solution to Example 15Right-hand side:Definition of field B on segments ab, bc, cd, and da is
*UNIVERSITI MALAYSIA PERLIS
Solution to Example 15At different segments,
Thus,
which is the same as the left hand side (proved!)
*UNIVERSITI MALAYSIA PERLIS
Laplacian OperatorLaplacian of a Scalar FieldLaplacian of a Vector Field
Useful Relation
Laplacian of a ScalarLaplacian of a scalar V is denoted by .
The result is a scalar.*UNIVERSITI MALAYSIA PERLIS
Laplacian Cylindrical
Laplacian Spherical
*
Example 16Find the Laplacian of these scalars:
*(a)(b)(c)UNIVERSITI MALAYSIA PERLIS
Solution to Example 16(a)
(b)
(c) *UNIVERSITI MALAYSIA PERLIS
Laplacian of a vectorFor vector E given in Cartesian coordinates as:
the Laplacian of vector E is defined as:
*UNIVERSITI MALAYSIA PERLIS
Laplacian of a vectorIn Cartesian coordinates, the Laplacian of a vector is a vector whose components are equal to the Laplacians of the vector components.Through direct substitution, we can simplify it as*UNIVERSITI MALAYSIA PERLIS
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