Ch 1 vectors ABEER ALSHAMMARI 1 Electromagnetism Introduction

Ch 1: vectors ABEER ALSHAMMARI 1

Electromagnetism Introduction to Electrodynamics Fourth Edition David J. Griths ABEER ALSHAMMARI 2

Some Definitions Scalar quantities: Quantities having magnitude only. Length, mass, time, temperature, energy. Vector quantities: Quantities having both magnitude and directions. Displacement, velocity, acceleration, momentum, angular momentum, electric field, magnetic field, dipoles. Tensor quantities: (Tensors of rank n) moment of inertia, electric permittivity, nonlinear susceptibility. Geometrical representation of a vector: An arrow. A ABEER ALSHAMMARI 3

Vector Math Vector Inverse ◦ Just switch direction Vector Addition ◦ Use head-tail method, or parallelogram method Vector Subtraction ◦ Use inverse, then add Vector Math by Components Vector Multiplication ◦ ◦ Three kinds! Multiplying a vector by a scalar Scalar, or dot product Vector, or cross product February 18, 2011

Scalar Product of Two Vectors The scalar product of two vectors is written as ◦ It is also called the dot product ◦ q is the angle between A and B Applied to work, this means ` February 18, 2011

Dot Product The dot product says something about how parallel two vectors are. The dot product (scalar product) of two vectors can be thought of as the projection of one onto the direction of the other. q Components February 18, 2011

Projection of a Vector: Dot Product The dot product says something about how parallel two vectors are. The dot product (scalar product) of two vectors can be thought of as the projection of one onto the direction of the other. Components Dot product is Commutative? Projection is zero p/2 Distributive? February 18, 2011

Cross Product The cross product of two vectors says something about how perpendicular they are. q Magnitude: y ◦ ◦ is smaller angle between the vectors Cross product of any parallel vectors = zero Cross product is maximum for perpendicular vectors Cross products of Cartesian unit vectors: j i x k z i j k February 18, 2011

Cross Product Direction: C perpendicular to both A and B (righthand rule) ◦ ◦ Place A and B tail to tail Right hand, not left hand Four fingers are pointed along the first vector A “sweep” from first vector A into second vector B through the smaller angle between them ◦ Your outstretched thumb points the direction of C First practice February 18, 2011

More about Cross Product The quantity ABsin is the area of the parallelogram formed by A and B The direction of C is perpendicular to the plane formed by A and B Cross product is not commutative The distributive law The derivative of cross product obeys the chain rule Calculate cross product February 18, 2011

Calculating Cross Products Find: Where: Solution: i j Calculate torque given a force and its location Solution: February 18, 2011 k

Triple Product 1 - Scalar triple product Geometrically, it is the volume of the parallelepiped generated by A, B and C 2 - vector triple product ABEER ALSHAMMARI 12

Ordinary derivative It tells us how the function f(x) rabidly varies when we change x by tiny amount dx Geometrical Interpretation: The derivative is the slope of the graph of f versus x Time derivative ABEER ALSHAMMARI 13

The gradient (scalar to vector) For a function T(x, y, z) ABEER ALSHAMMARI 14

Suppose and is the temperature at , is the temperature at as shown. The differential distances are the components of the differential distance vector : However, from differential calculus, the differential temperature: ABEER ALSHAMMARI 15

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. ABEER ALSHAMMARI 16

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Example: ABEER ALSHAMMARI 18

The Divergence (vector to scalar) The divergence of vector field is a measure of how the vector spreads out (diverges) from a specific point. Also it describes the net flux of the vector field leaving or entering a small volume within the field Divergence of scalar is meaningless What does positive, negative, zero divergence mean? ABEER ALSHAMMARI 19

Example Find the divergence of ABEER ALSHAMMARI 20

The Curl The curl of a vector field is the area density of the circulation of the field(circulation per unit area). It is a measure of how the vector curls around the point. Curl of scalar is meaningless ABEER ALSHAMMARI 21

Example Find the curl of ABEER ALSHAMMARI 22

The Laplacian measures the difference between the average value of the field around a point and the value of the field at that point. If then j cannot increase or decrease in all directions. The Laplacian is analogous to the second derivative Tells us the concavity of the scalar field in 3 D. How the change is changing from point to another point. ABEER ALSHAMMARI 23

Example Calculate the laplacian of ABEER ALSHAMMARI 24

Problems Proof that Proof ABEER ALSHAMMARI 25

Line, Surface, Volume Integrals a) line integral:

Example 1. 6

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b) surface integral: For a given boundary line there many different surfaces, on which the surface integral depends. It is independent only if If the surface is closed:

Example 1. 7 2 2 2

volume integral: Example 1. 8