APPLICATION OFVECTOR ANALYSIS

Alipar, Emmanuel B.BSECE - 3

C O N T E N T S

I. What is Vector?

II. Application of Vector AnalysisDivergence of Vector Field

Position Vector

Vector Analysis

it has a quantity that has direction as well as magnitude

Tail

Tip / Arrowhead

Vector Analysis

DOT PRODUCTCommutative: A + B = B + A

Associative: A + (B + C) = (A + B)=C

Distributive: K (A + B)=KA + KB

CROSS PRODUCTAnti Commutative: A x B = B x A but A x B = -B x A

It is not an Associative: A x (B x C) = (A x B) x C

It is a Distributive: A x (B + C) = A x B + A x C

1. Electromagnetism (Gauss's Law)

is a law relating the distribution of electric charge to the resulting electric field. Gauss's law states that: electric flux through any closed surface is proportional to the enclosed electric charge.

provides a relationship between the net electrical flux through a closed surface and the net charge enclosed by that surface. Gauss law states that the net electric flux out of a closed surface is equal to the charge enclosed by the surface divided by the permittivity.

Combination of Gauss's Law &Divergence of Vector field

Electromagnetic Application (Gauss's Law)

In Electromagnetism, one of its application is the Gauss's Law which is combined with divergence theorem and surface integrals of a vector field.

Electromagnetic Application (Gauss's Law)

Electromagnetic Application (Gauss's Law)

continuation...

Divergence of the Vector Fields

is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

Divergence of the vector field

* The direction of the arrow indicates the flow source of outward and inward flux.

2. Signal Processing

it is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of analog as well as digitized signals, representing time-varying or spatially varying physical quantities.

Signals of interest can include sound, electromagnetic radiation, images, and sensor readings, for example biological measurements such as electrocardiograms, control system signals, telecommunication transmission signals, and many others.

Example of Signal ProcessingConsider the problem of designing a smooth takeoff path for an airplane. The initial vector is the plane heading down the runway just before takeoff. The final vector is the level flight path at a given altitude. You must design the path the plane takes between these two vectors while avoiding jarring the passengers too much during the takeoff.

Example of Signal Processing (Cont.)1. Idealization: We will idealize the aircraft as a particle. We can do this because the aircraft is not rotating during takeoff.2. FBD: The figure shows a free body diagram. represents the (unknown) force exerted on the aircraft due to its engines.3. Kinematics: We must calculate the acceleration required to reach takeoff speed. We are given (i) the distance to takeoff d, (ii) the takeoff speed and (iii) the aircraft is at rest at the start of the takeoff roll. We can therefore write down the position vector r and velocity v of the aircraft at takeoff, and use the straight line motion formulas for r and v to calculate the time t to reach takeoff speed and the acceleration a. Taking the origin at the initial position of the aircraft, we have that, at the instant of takeoff

Diagram Airplane Take-off Signal

Example of Signal Processing (Cont.)The Vector Transition plot shows one possible smoothed trajectory (in blue) compared to the direct path (in red). The blue trace seems sufficiently gradual to make a good choice. How was this path computed? Look at the Second Derivative (Acceleration) plot, which has two back-to-back Hann windows (hann) .

Position Vector

The position vector of point P is the directed distance from the origin O to P.

Formula:rP = OP =xax + yay + zaz

rPQ = rQ rP= (xQ xP)ax + (yQ yP)ay + (zQ - zP)az

Illustration of Position Vector

z

y

P(1,2,3)

x

rP = xax + yay + zaz