ekt 241-2- vector analysis 2013 dr ruzelita

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


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.


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.


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

Coordinate SystemsCartesian coordinates

Circular Cylindrical coordinates

Spherical coordinates


Cartesian coordinatesConsists of three mutually orthogonal axes and a point in space is denoted as


Cartesian CoordinatesUnit vector of in the direction of increasing coordinate value.


Cartesian CoordinatesDifferential in Length


Cartesian CoordinatesDifferential Surface*UNIVERSITI MALAYSIA PERLIS

Cartesian CoordinatesDifferential Surface


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.


*Differential in LengthCircular Cylindrical CoordinatesUNIVERSITI MALAYSIA PERLIS

Circular Cylindrical CoordinatesIncrement in length for direction is:

is not increment in length!


Circular Cylindrical CoordinatesDifferential Surface*UNIVERSITI MALAYSIA PERLIS

Circular Cylindrical CoordinatesDifferential volume*UNIVERSITI MALAYSIA PERLIS

Calculus Basic

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.


FORMULADifferential volume*UNIVERSITI MALAYSIA PERLIS

Solution to Example 6 (i) For volume enclosed, we integrate;


*Differential SurfaceFORMULAUNIVERSITI MALAYSIA PERLIS

Solution to Example 6 (ii) For surface area, we add the area of each surfaces;


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

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


Spherical CoordinatesUnit vector of in the direction of increasing coordinate value.


Spherical CoordinatesDifferential in length


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.


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


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,


Example 11*Find the directional derivative of

along the direction and evaluate it at (1,1, 2).


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:


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


Divergence of a vector field*Net outward flux per unit volume i.e the div of E is


Example 12Find divergence of these vectors:

*(a)

(b)


Solution to Example 12(a) Use divergence for Cartesian coordinate:*UNIVERSITI MALAYSIA PERLIS

Solution to Example 12(b) Use divergence for cylindrical coordinate:


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


Curl of a vector fieldNon-uniform field, e.g azimuthal field, circulation is not zero


Vector identities involving curlFor any two vectors A and B:


Curl in Cartesian coordinatesFor Cartesian coordinates:


Curl in cylindrical coordinatesFor cylindrical coordinates:


Curl in spherical coordinatesFor spherical coordinates:*UNIVERSITI MALAYSIA PERLIS

Example 14Find curl of these vectors:

*(a)

(b)


Solution to Example 14(a) Use curl for Cartesian coordinate:


Solution to Example 14(b) Use curl for cylindrical coordinate


Solution to Example 14(c) Use curl for Spherical coordinate:


Solution to Example 14*UNIVERSITI MALAYSIA PERLIS

Solution to Example 14(c) continued


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.


Example 15*UNIVERSITI MALAYSIA PERLIS

Solution to Example 15Stokess theorem states that:

Left-hand side: First, use curl in cylindrical coordinates


Solution to Example 15The integral of over the specified surface S with r = 2 is:


Solution to Example 15Right-hand side:Definition of field B on segments ab, bc, cd, and da is


Solution to Example 15At different segments,

Thus,

which is the same as the left hand side (proved!)


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:


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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ekt 241-2- vector analysis 2013 dr ruzelita

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