Soh Ping Jack, Azremi Abdullah Al-Hadi, Ruzelita Ngadiran 3. Vector Analysis Soh Ping Jack, Azremi Abdullah Al-Hadi, Ruzelita Ngadiran

Overview Basic Laws of Vector Algebra Dot Product and Cross Product Orthogonal Coordinate Systems: Cartesian, Cylindrical and Spherical Coordinate Systems Transformations between Coordinate Systems Gradient of a Scalar Field Divergence of a Vector Field Divergence Theorem Curl of a Vector Field Stokes’s Theorem Laplacian Operator

This chapter cover CO1 Ability to describe different coordinate system and their interrelation.

Scalar A scalar is a quantity that has only magnitude E.g. of Scalars: Time, mass, distance, temperature, electrical potential etc

Vector A 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 Algebra Cartesian coordinate systems

Vector in Cartesian Coordinates A vector in Cartesian Coordinates maybe represented as OR

Vector in Cartesian Coordinates Vector 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 Unit vector magnitude magnitude Unit vector

Example 1 : Unit Vector Specify 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 Operations Equality of Two Vectors

Vector Algebra For addition and subtraction of A and B, Hence, Commutative property

Example 2 : If Find: (a) The component of along (b) The magnitude of (c) A unit vector along

Solution to Example 2 (a) The component of along is (b)

Cont Hence, the magnitude of is: (c) Let

Cont So, the unit vector along is:

Position & Distance Vectors Position Vector: From origin to point P Distance Vector: Between two points

Position and distance Vector

Example 3 Point 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

UNIVERSITI MALAYSIA PERLIS Solution to Example 3 (a) (b) (c) 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 UNIVERSITI MALAYSIA PERLIS Since is a distance vector, the distance between P and Q is the magnitude of this distance vector.

UNIVERSITI MALAYSIA PERLIS Solution to Example 3 Distance, d UNIVERSITI MALAYSIA PERLIS (d) Let the required vector be then Where is the magnitude of

UNIVERSITI MALAYSIA PERLIS Solution to Example 3 Since is parallel to , it must have the same unit vector as or UNIVERSITI MALAYSIA PERLIS So,

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) product UNIVERSITI MALAYSIA PERLIS

UNIVERSITI 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: UNIVERSITI MALAYSIA PERLIS And thus, only three terms remain, giving finally:

Dot Product in Cartesian The two vectors, and are said to be perpendicular or orthogonal (90°) with each other if; UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Laws of Dot Product Dot 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 them UNIVERSITI MALAYSIA PERLIS

Vector or Cross Product Direction of is perpendicular (90°) to the plane containing A and B

Vector or Cross Product It 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 gives UNIVERSITI MALAYSIA PERLIS and 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 distributive UNIVERSITI MALAYSIA PERLIS

Properties of Vector Product Properties of cross product of unit vectors: UNIVERSITI MALAYSIA PERLIS Or by using cyclic permutation:

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

UNIVERSITI MALAYSIA PERLIS Solution to Example 4 The dot product is: UNIVERSITI MALAYSIA PERLIS

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 Products Scalar Triple Product Vector Triple Product

UNIVERSITI MALAYSIA PERLIS Example 5 Given , and . Find (A×B)×C and compare it with A×(B×C). UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Solution to Example 5 A similar procedure gives UNIVERSITI MALAYSIA PERLIS

Cont’ Hence :

Example From Book Scalar/ dot product

Solution

Solution

Cont’

UNIVERSITI MALAYSIA PERLIS Coordinate Systems Cartesian coordinates Circular Cylindrical coordinates Spherical coordinates UNIVERSITI MALAYSIA PERLIS

Cartesian coordinates Consists of three mutually orthogonal axes and a point in space is denoted as UNIVERSITI MALAYSIA PERLIS

Cartesian Coordinates Unit vector of in the direction of increasing coordinate value. UNIVERSITI MALAYSIA PERLIS

Cartesian Coordinates Differential in Length UNIVERSITI MALAYSIA PERLIS

Cartesian Coordinates Differential Surface UNIVERSITI MALAYSIA PERLIS

Cartesian Coordinates Differential Surface UNIVERSITI MALAYSIA PERLIS

Cartesian Coordinates Differential Volume x y z UNIVERSITI MALAYSIA PERLIS

Cartesian Coordinate System Differential length vector Differential area vectors

Circular Cylindrical Coordinates x y z UNIVERSITI MALAYSIA PERLIS

Circular Cylindrical Coordinates Form 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

Circular Cylindrical Coordinates Differential in Length UNIVERSITI MALAYSIA PERLIS

Circular Cylindrical Coordinates Increment in length for direction is: is not increment in length! UNIVERSITI MALAYSIA PERLIS

Circular Cylindrical Coordinates Differential Surface UNIVERSITI MALAYSIA PERLIS

Circular Cylindrical Coordinates Differential volume UNIVERSITI MALAYSIA PERLIS

Cylindrical Coordinate System

Cylindrical Coordinate System

UNIVERSITI MALAYSIA PERLIS 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

UNIVERSITI MALAYSIA PERLIS FORMULA Differential volume UNIVERSITI MALAYSIA PERLIS

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

UNIVERSITI MALAYSIA PERLIS FORMULA Differential Surface UNIVERSITI MALAYSIA PERLIS 72

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

UNIVERSITI MALAYSIA PERLIS Example 7 The surfaces define a closed surface. Find: The enclosed volume. The total area of the enclosing surface. UNIVERSITI MALAYSIA PERLIS To answer (b) look formula slide 77. The cylindrical form is same as slide 77

UNIVERSITI MALAYSIA PERLIS Solution to Example 7 (a) The enclosed volume; UNIVERSITI MALAYSIA PERLIS Must convert into radians

UNIVERSITI MALAYSIA PERLIS FORMULA Differential Surface UNIVERSITI MALAYSIA PERLIS 76

answer (b): Draw the diagram Differential Surface

UNIVERSITI MALAYSIA PERLIS Solution to Example 7 (b) The total area of the enclosed surface: With radius of 3: Top and Down: UNIVERSITI MALAYSIA PERLIS With radius of 5: There 2 Curve area:

Remember This Figure of cylindrical coordinate EXERCISE Intergral the highligted surface only . The highligt are is perpendicular to the dsr ONLY!! So intergrate dsr ONLY!! Remember This Figure of cylindrical coordinate

From Book

From Book

Spherical Coordinates UNIVERSITI MALAYSIA PERLIS

Spherical Coordinates Point 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 Coordinates Unit vector of in the direction of increasing coordinate value. UNIVERSITI MALAYSIA PERLIS

Spherical Coordinates Differential in length UNIVERSITI MALAYSIA PERLIS

Spherical Coordinates Differential Surface UNIVERSITI MALAYSIA PERLIS

Spherical Coordinates Differential Surface UNIVERSITI MALAYSIA PERLIS

Spherical Coordinates Differential Volume UNIVERSITI MALAYSIA PERLIS

Spherical Coordinates However, 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

UNIVERSITI MALAYSIA PERLIS Example 8a A sphere of radius 2 cm contains a volume charge density ρv given by; Find the total charge Q contained in the sphere. UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Solution: Example 8a UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Example 8b The spherical strip is a section of a sphere of radius 3 cm. Find the area of the strip. UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Solution : Example 8b Use the elemental area with constant R, that is . Solution: UNIVERSITI MALAYSIA PERLIS

Exercise Answer

Coordinate Transformations: Coordinates To solve a problem, we select the coordinate system that best fits its geometry Sometimes we need to transform between coordinate systems

Coordinate Transformations: Unit Vectors

Cartesian to Cylindrical Transformations Relationships between (x, y, z) and (r, φ, z) are shown. UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Cartesian to Spherical Transformations Relationships between (x, y, z) and (r, θ, Φ) are shown in the diagram. UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Cartesian to Spherical Transformations Relationships between (x, y, z) and (r, θ, Φ) are shown. UNIVERSITI MALAYSIA PERLIS

Example 9 Solution Express vector in spherical coordinates. Using the transformation relation, Using the expressions for x, y, and z, Solution

UNIVERSITI MALAYSIA PERLIS Example 9: contd Similarly, substituting the expression for x, y, z for; we get: Hence, UNIVERSITI MALAYSIA PERLIS

Ex: Cartesian to Cylindrical Φ in degree

Distance Between 2 Points

UNIVERSITI MALAYSIA PERLIS Transformations Distance d between two points is Converting to cylindrical equivalents Converting to spherical equivalents UNIVERSITI MALAYSIA PERLIS

VECTOR CALCULUS 1 GRADIENT OF A SCALAR 2 DIVERGENCE OF A VECTOR 3 DIVERGENCE THEOREM 4 CURL OF A VECTOR 5 STOKES’S THEOREM 6 LAPLACIAN OF A SCALAR

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 : UNIVERSITI MALAYSIA PERLIS However, from differential calculus, the differential temperature:

Gradient of a scalar field But, UNIVERSITI MALAYSIA PERLIS So, previous equation can be rewritten as:

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. UNIVERSITI MALAYSIA PERLIS For Cartesian coordinate, grad T: The symbol is called the del or gradient operator.

Gradient operator in cylindrical and spherical coordinates Gradient operator in cylindrical coordinates: Gradient operator in spherical coordinates: UNIVERSITI MALAYSIA PERLIS After this, Go to slide 115

Gradient of A Scalar Field

Gradient ( cont.)

UNIVERSITI MALAYSIA PERLIS Example 10 Find the gradient of these scalars: (a) (b) (c) UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Solution to Example 10 (a) Use gradient for Cartesian coordinate: UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Solution to Example 10 (b) Use gradient for cylindrical coordinate: UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Solution to Example 10 (c) Use gradient for Spherical coordinate: UNIVERSITI MALAYSIA PERLIS

Directional derivative Gradient 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

UNIVERSITI MALAYSIA PERLIS Example 11 Find the directional derivative of along the direction and evaluate it at (1,−1, 2). UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Solution to Example 11 GradT : We denote L as the given direction, Unit vector is and UNIVERSITI MALAYSIA PERLIS

Divergence of a vector field Illustration of the divergence of a vector field at point P: UNIVERSITI MALAYSIA PERLIS Positive Divergence Negative Divergence Zero Divergence

Divergence of a vector field The divergence of A at a given point P is the net outward flux per unit volume: UNIVERSITI MALAYSIA PERLIS

Divergence of a vector field Vector field A at closed surface S What is ?? UNIVERSITI MALAYSIA PERLIS

Divergence of a vector field Where, UNIVERSITI MALAYSIA PERLIS And, v is volume enclosed by surface S

Divergence of a vector field For Cartesian coordinate: UNIVERSITI MALAYSIA PERLIS For Circular cylindrical coordinate:

Divergence of a vector field For Spherical coordinate: UNIVERSITI MALAYSIA PERLIS

Divergence of a vector field Example: A point charge q Total 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

UNIVERSITI MALAYSIA PERLIS Example 12 Find divergence of these vectors: (a) (b) (c) UNIVERSITI MALAYSIA PERLIS

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

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

UNIVERSITI MALAYSIA PERLIS Solution to Example 12 (c) Use divergence for Spherical coordinate: UNIVERSITI MALAYSIA PERLIS

Divergence of a Vector Field

Divergence Theorem Useful tool for converting integration over a volume to one over the surface enclosing that volume, and vice versa

Curl of a Vector Field

UNIVERSITI MALAYSIA PERLIS Curl of a vector field The 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

UNIVERSITI MALAYSIA PERLIS Curl of a vector field The circulation of B around closed contour C: UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Curl of a vector field Curl of a vector field B is defined as: UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Curl of a vector field Curl is used to measure the uniformity of a field Uniform field, circulation is zero Non-uniform field, e.g azimuthal field, circulation is not zero UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Curl of a vector field Uniform field, circulation is zero UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Curl of a vector field Non-uniform field, e.g azimuthal field, circulation is not zero UNIVERSITI MALAYSIA PERLIS

Vector identities involving curl For any two vectors A and B: UNIVERSITI MALAYSIA PERLIS

Curl in Cartesian coordinates For Cartesian coordinates: Look Component az and differentiate against y UNIVERSITI MALAYSIA PERLIS

Curl in cylindrical coordinates For cylindrical coordinates: UNIVERSITI MALAYSIA PERLIS

Curl in spherical coordinates For spherical coordinates: UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Example 14 Find curl of these vectors: (a) (b) (c) UNIVERSITI MALAYSIA PERLIS

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

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

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

UNIVERSITI MALAYSIA PERLIS Solution to Example 14 UNIVERSITI MALAYSIA PERLIS

Solution to Example 14 stop (c) continued… UNIVERSITI MALAYSIA PERLIS

Stokes’s Theorem

UNIVERSITI MALAYSIA PERLIS Stokes’s Theorem Converts 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

UNIVERSITI MALAYSIA PERLIS Example 15 A vector field is given by . Verify Stokes’s theorem for a segment of a cylindrical surface defined by r = 2, π/3 ≤ φ ≤ π/2, 0 ≤ z ≤ 3 as shown in the diagram on the next slide. UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Example 15 UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Solution to Example 15 Stokes’s theorem states that: Left-hand side: First, use curl in cylindrical coordinates UNIVERSITI MALAYSIA PERLIS

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

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

UNIVERSITI MALAYSIA PERLIS Solution to Example 15 At different segments, Thus, which is the same as the left hand side (proved!) UNIVERSITI MALAYSIA PERLIS

Laplacian Operator Laplacian of a Scalar Field Laplacian of a Vector Field Useful Relation

UNIVERSITI MALAYSIA PERLIS Laplacian of a Scalar Laplacian of a scalar V is denoted by . The result is a scalar. UNIVERSITI MALAYSIA PERLIS

Laplacian Cylindrical Laplacian Spherical

UNIVERSITI MALAYSIA PERLIS Example 16 Find the Laplacian of these scalars: (a) UNIVERSITI MALAYSIA PERLIS (b) (c)

UNIVERSITI MALAYSIA PERLIS Solution to Example 16 (a) (b) (c) UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Laplacian of a vector For vector E given in Cartesian coordinates as: the Laplacian of vector E is defined as: UNIVERSITI MALAYSIA PERLIS

UNIVERSITI MALAYSIA PERLIS Laplacian of a vector In 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