EKT 241-2- Vector Analysis 2013 DR Ruzelita
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Transcript of EKT 241-2- Vector Analysis 2013 DR Ruzelita
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3. VECTOR ANALYSISRuzelita Ngadiran
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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
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This chapter cover CO1Ability to describe different coordinate system and their interrelation.
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ScalarA scalar is a quantity that has only magnitudeE.g. of Scalars:Time, mass, distance, temperature, electrical potential etc
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VectorA vector is a quantity that has both magnitude and direction.E.g. of Vectors:Velocity, force, displacement, electric field intensity etc.
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Basic Laws of Vector AlgebraCartesian coordinate systems
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Vector in Cartesian CoordinatesA vector in Cartesian Coordinates maybe represented as
OR
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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.
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Laws of Vector Algebra stopmagnitudeUnit vectormagnitudeUnit vector
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Example 1 : Unit VectorSpecify the unit vector extending from the origin towards the point
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Solution :Construct the vector extending from origin to point G
Find the magnitude of
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Solution :So, unit vector is
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Properties of Vector OperationsEquality of Two Vectors
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Vector AlgebraFor addition and subtraction of A and B,Hence,Commutative property
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Example 2 : If
Find:
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Solution to Example 2(a)The component of along is (b)
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ContHence, the magnitude of is:(c) Let
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ContSo, the unit vector along is:
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Position & Distance VectorsPosition Vector: From origin to point PDistance Vector: Between two points
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Position and distance Vector
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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
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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
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Solution to Example 3*Distance, d (d)
Let the required vector be thenWhere is the magnitude of UNIVERSITI MALAYSIA PERLIS
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Solution to Example 3*Since is parallel to , it must have the same unit vector as orSo, UNIVERSITI MALAYSIA PERLIS
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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
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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
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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
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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
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Dot Product in CartesianThe two vectors, and are said to be perpendicular or orthogonal (90) with each other if;
*UNIVERSITI MALAYSIA PERLIS
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Laws of Dot ProductDot product obeys the following:
Commutative Law
Distributive Law
*UNIVERSITI MALAYSIA PERLIS
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Properties of dot product*
Properties of dot product of unit vectors:UNIVERSITI MALAYSIA PERLIS
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Vector Multiplication: Scalar Product or Dot Product
Hence:
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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
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Vector or Cross Product*Direction of is perpendicular (90) to the plane containing A and B
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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
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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
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Cross product in Cartesian* By using the properties of cross product, it givesand be written in more easily remembered form:UNIVERSITI MALAYSIA PERLIS
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Laws of Vector Product* Cross product obeys the following: It is not commutative It is not associative It is distributiveUNIVERSITI MALAYSIA PERLIS
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Properties of Vector Product*Properties of cross product of unit vectors:Or by using cyclic permutation:UNIVERSITI MALAYSIA PERLIS
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Vector Multiplication: Vector Product or Cross Product
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Example 4:Dot & Cross Product*Determine the dot product and cross product of the following vectors:UNIVERSITI MALAYSIA PERLIS
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Solution to Example 4*The dot product is:UNIVERSITI MALAYSIA PERLIS
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Solution to Example 4*The cross product is:UNIVERSITI MALAYSIA PERLIS
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Scalar & Vector Triple Product A scalar triple product is
A vector triple product is
known as the bac-cab rule.
*UNIVERSITI MALAYSIA PERLIS
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Triple ProductsScalar Triple ProductVector Triple Product
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Example 5
Given , and .
Find (AB)C and compare it with A(BC). *UNIVERSITI MALAYSIA PERLIS
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Solution to Example 5
A similar procedure gives *UNIVERSITI MALAYSIA PERLIS
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ContHence :
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Example From Book Scalar/ dot product
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Solution
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Solution
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Cont
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Coordinate SystemsCartesian coordinates
Circular Cylindrical coordinates
Spherical coordinates
*UNIVERSITI MALAYSIA PERLIS
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Cartesian coordinatesConsists of three mutually orthogonal axes and a point in space is denoted as
*UNIVERSITI MALAYSIA PERLIS
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Cartesian CoordinatesUnit vector of in the direction of increasing coordinate value.
*UNIVERSITI MALAYSIA PERLIS
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Cartesian CoordinatesDifferential in Length
*UNIVERSITI MALAYSIA PERLIS
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Cartesian CoordinatesDifferential Surface*UNIVERSITI MALAYSIA PERLIS
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Cartesian CoordinatesDifferential Surface
*UNIVERSITI MALAYSIA PERLIS
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Cartesian CoordinatesDifferential Volume*UNIVERSITI MALAYSIA PERLIS
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Cartesian Coordinate SystemDifferential length vectorDifferential area vectors
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Circular Cylindrical Coordinates*UNIVERSITI MALAYSIA PERLIS
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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
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*Differential in LengthCircular Cylindrical CoordinatesUNIVERSITI MALAYSIA PERLIS
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Circular Cylindrical CoordinatesIncrement in length for direction is:
is not increment in length!
*UNIVERSITI MALAYSIA PERLIS
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Circular Cylindrical CoordinatesDifferential Surface*UNIVERSITI MALAYSIA PERLIS
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Circular Cylindrical CoordinatesDifferential volume*UNIVERSITI MALAYSIA PERLIS
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Calculus Basic
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Cylindrical Coordinate System
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Cylindrical Coordinate System
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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
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FORMULADifferential volume*UNIVERSITI MALAYSIA PERLIS
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Solution to Example 6 (i) For volume enclosed, we integrate;
*UNIVERSITI MALAYSIA PERLIS
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*Differential SurfaceFORMULAUNIVERSITI MALAYSIA PERLIS
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Solution to Example 6 (ii) For surface area, we add the area of each surfaces;
*UNIVERSITI MALAYSIA PERLIS
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Example 7The surfaces define a closed surface. Find:
The enclosed volume. The total area of the enclosing surface.*UNIVERSITI MALAYSIA PERLIS
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Solution to Example 7(a) The enclosed volume;
*Must convert into radiansUNIVERSITI MALAYSIA PERLIS
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Solution to Example 7(b) The total area of the enclosed surface:*UNIVERSITI MALAYSIA PERLIS
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From Book
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From Book
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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
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Spherical Coordinates*UNIVERSITI MALAYSIA PERLIS
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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
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Spherical CoordinatesUnit vector of in the direction of increasing coordinate value.
*UNIVERSITI MALAYSIA PERLIS
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Spherical CoordinatesDifferential in length
*UNIVERSITI MALAYSIA PERLIS
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Spherical CoordinatesDifferential Surface *UNIVERSITI MALAYSIA PERLIS
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Spherical CoordinatesDifferential Surface*UNIVERSITI MALAYSIA PERLIS
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Spherical CoordinatesDifferential Volume*UNIVERSITI MALAYSIA PERLIS
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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
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Spherical Coordinate System
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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
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Solution: Example 8a*UNIVERSITI MALAYSIA PERLIS
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Example 8bThe spherical strip is a section of a sphere of radius 3 cm. Find the area of the strip.
*UNIVERSITI MALAYSIA PERLIS
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Solution : Example 8bUse the elemental area with constant R, that is . Solution:*UNIVERSITI MALAYSIA PERLIS
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ExerciseAnswer
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Coordinate Transformations: CoordinatesTo solve a problem, we select the coordinate system that best fits its geometrySometimes we need to transform between coordinate systems
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Coordinate Transformations: Unit Vectors
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Cartesian to Cylindrical Transformations*Relationships between (x, y, z) and (r, , z) are shown.UNIVERSITI MALAYSIA PERLIS
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*Cartesian to Spherical Transformations
Relationships between (x, y, z) and (r, , ) are shown in the diagram.UNIVERSITI MALAYSIA PERLIS
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*Cartesian to Spherical Transformations
Relationships between (x, y, z) and (r, , ) are shown.UNIVERSITI MALAYSIA PERLIS
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Example 9*Express vector in spherical coordinates.
Using the transformation relation,
Using the expressions for x, y, and z,Solution
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Example 9: contd*Similarly, substituting the expression for x, y, z for;
we get:
Hence,UNIVERSITI MALAYSIA PERLIS
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Ex: Cartesian to Cylindrical in degree
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Distance Between 2 Points
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TransformationsDistance d between two points is
Converting to cylindrical equivalents
Converting to spherical equivalents
*UNIVERSITI MALAYSIA PERLIS
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Gradient of a scalar field*Suppose is the temperature at ,and is the temperature at as shown. UNIVERSITI MALAYSIA PERLIS
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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
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Gradient of a scalar field*But,So, previous equation can be rewritten as:UNIVERSITI MALAYSIA PERLIS
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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
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Gradient operator in cylindrical and spherical coordinatesGradient operator in cylindrical coordinates:
Gradient operator in spherical coordinates:
*UNIVERSITI MALAYSIA PERLISAfter this, Go to slide 115
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Gradient of A Scalar Field
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Gradient ( cont.)
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Example 10Find the gradient of these scalars:*(a)
(b)
(c)UNIVERSITI MALAYSIA PERLIS
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Solution to Example 10*(a) Use gradient for Cartesian coordinate:UNIVERSITI MALAYSIA PERLIS
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Solution to Example 10*(b) Use gradient for cylindrical coordinate:UNIVERSITI MALAYSIA PERLIS
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Solution to Example 10*(c) Use gradient for Spherical coordinate:UNIVERSITI MALAYSIA PERLIS
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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
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Example 11*Find the directional derivative of
along the direction and evaluate it at (1,1, 2).
UNIVERSITI MALAYSIA PERLIS
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Solution to Example 11*
GradT :
We denote L as the given direction,
Unit vector is
andUNIVERSITI MALAYSIA PERLIS
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Divergence of a vector fieldIllustration of the divergence of a vector field at point P:
*Positive DivergenceNegative DivergenceZero DivergenceUNIVERSITI MALAYSIA PERLIS
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Divergence of a vector fieldThe divergence of A at a given point P is the net outward flux per unit volume:
*UNIVERSITI MALAYSIA PERLIS
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Divergence of a vector field*What is ?? Vector field A at closed surface SUNIVERSITI MALAYSIA PERLIS
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Divergence of a vector field*Where,And, v is volume enclosed by surface SUNIVERSITI MALAYSIA PERLIS
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Divergence of a vector field*For Cartesian coordinate:For Circular cylindrical coordinate:UNIVERSITI MALAYSIA PERLIS
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Divergence of a vector field*For Spherical coordinate:UNIVERSITI MALAYSIA PERLIS
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Divergence of a vector field Tahun 4Example: A point charge qTotal flux of the electric field E due to q is
*UNIVERSITI MALAYSIA PERLIS
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Divergence of a vector field*Net outward flux per unit volume i.e the div of E is
UNIVERSITI MALAYSIA PERLIS
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Example 12Find divergence of these vectors:
*(a)
(b)
(c)UNIVERSITI MALAYSIA PERLIS
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Solution to Example 12(a) Use divergence for Cartesian coordinate:*UNIVERSITI MALAYSIA PERLIS
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Solution to Example 12(b) Use divergence for cylindrical coordinate:
*UNIVERSITI MALAYSIA PERLIS
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Solution to Example 12(c) Use divergence for Spherical coordinate:*UNIVERSITI MALAYSIA PERLIS
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Divergence of a Vector Field
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Divergence TheoremUseful tool for converting integration over a volume to one over the surface enclosing that volume, and vice versa
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Curl of a Vector Field
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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
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Curl of a vector fieldThe circulation of B around closed contour C:*UNIVERSITI MALAYSIA PERLIS
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Curl of a vector fieldCurl of a vector field B is defined as:*UNIVERSITI MALAYSIA PERLIS
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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
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Curl of a vector fieldUniform field, circulation is zero
*UNIVERSITI MALAYSIA PERLIS
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Curl of a vector fieldNon-uniform field, e.g azimuthal field, circulation is not zero
*UNIVERSITI MALAYSIA PERLIS
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Vector identities involving curlFor any two vectors A and B:
*UNIVERSITI MALAYSIA PERLIS
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Curl in Cartesian coordinatesFor Cartesian coordinates:
*UNIVERSITI MALAYSIA PERLIS
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Curl in cylindrical coordinatesFor cylindrical coordinates:
*UNIVERSITI MALAYSIA PERLIS
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Curl in spherical coordinatesFor spherical coordinates:*UNIVERSITI MALAYSIA PERLIS
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Example 14Find curl of these vectors:
*(a)
(b)
(c)UNIVERSITI MALAYSIA PERLIS
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Solution to Example 14(a) Use curl for Cartesian coordinate:
*UNIVERSITI MALAYSIA PERLIS
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Solution to Example 14(b) Use curl for cylindrical coordinate
*UNIVERSITI MALAYSIA PERLIS
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Solution to Example 14(c) Use curl for Spherical coordinate:
*UNIVERSITI MALAYSIA PERLIS
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Solution to Example 14*UNIVERSITI MALAYSIA PERLIS
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Solution to Example 14(c) continued
*UNIVERSITI MALAYSIA PERLIS
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Stokess Theorem
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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
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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
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Example 15*UNIVERSITI MALAYSIA PERLIS
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Solution to Example 15Stokess theorem states that:
Left-hand side: First, use curl in cylindrical coordinates
*UNIVERSITI MALAYSIA PERLIS
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Solution to Example 15The integral of over the specified surface S with r = 2 is:
*UNIVERSITI MALAYSIA PERLIS
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Solution to Example 15Right-hand side:Definition of field B on segments ab, bc, cd, and da is
*UNIVERSITI MALAYSIA PERLIS
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Solution to Example 15At different segments,
Thus,
which is the same as the left hand side (proved!)
*UNIVERSITI MALAYSIA PERLIS
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Laplacian OperatorLaplacian of a Scalar FieldLaplacian of a Vector Field
Useful Relation
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Laplacian of a ScalarLaplacian of a scalar V is denoted by .
The result is a scalar.*UNIVERSITI MALAYSIA PERLIS
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Laplacian Cylindrical
Laplacian Spherical
*
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Example 16Find the Laplacian of these scalars:
*(a)(b)(c)UNIVERSITI MALAYSIA PERLIS
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Solution to Example 16(a)
(b)
(c) *UNIVERSITI MALAYSIA PERLIS
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Laplacian of a vectorFor vector E given in Cartesian coordinates as:
the Laplacian of vector E is defined as:
*UNIVERSITI MALAYSIA PERLIS
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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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