8/18/2004 sec 2_2a.doc 1/2 Jim Stiles The Univ. of Kansas Dept. of EECS Chapter 2 –Vector Analysis 2-2 Physical Quantities and Units (pp. 7-11) A. Types of physical quantities HO: Examples of Physical Quantities B. Vector Representation8/18/2004 sec 2_2a.doc 2/2 Jim Stiles The Univ. of Kansas Dept. of EECS HO: Vector Representations C. The Directed Distance HO: The Directed Distance8/9/2004 Examples of Physical Quantities.doc 1/6 Jim Stiles The University of Kansas EECS Examples of Physical Quantities A. Discrete Scalar Quantities can be described with a single numeric value. Examples include: 1) My height (~ 6 ft.). 2) The weight of your text book (~ 1.0 lbs.) 3) The surface temperature of a specific location at a specific time. … indicating the surface temperature at a specific time and place! A discrete scalar quantity… Graphically, a discrete scalar quantity can be indicated as a point on a line, surface or volume, e.g.: T 100 Fo91 Fo 0 Fo8/9/2004 Examples of Physical Quantities.doc 2/6 Jim Stiles The University of Kansas EECS B. Discrete Vector Quantities must be described with both a magnitude and a direction. Examples include: 1) The force I am exerting on the floor (180 lbs. +++, in a direction toward the center of the earth). 2) The wind velocity of a specific location at a specific time. We will find that a discrete vector can be graphically represented as an arrow: Discrete scalar quantities. A discrete vector quantity! The wind has a magnitude of 8 mph and is blowing from the southwest (its direction). 14 mph North 23 mph West A specific place and time! wherein the length of the arrow is proportional to the magnitude and the orientation indicates direction.8/9/2004 Examples of Physical Quantities.doc 3/6 Jim Stiles The University of Kansas EECS C. Scalar Fields are quantities that must be described as one function of (typically) space and/or time. For example: 1) My weight as a function of time. Note that we cannot specify this as a single numerical value, as my weight has changed significantly over the course of my life! Instead, we must use a function of time to describe my weight: 2() 5.2 10 0.12Wtt=+ −t lbs. where t is my age in years. We can likewise graphically represent this scalar field by plotting the function W(t): Note that we can use this scalar field to determine discrete scalar values! For example, say we wish to determine my weight at birth. This is a discrete scalar value—it can be expressed numerically: 10 20 30 4050100150200 W(t) t 2( 0) 5.2 10(0) 0.12 (0) 5.2 lbs.Wt== + −= Why I’m always hungry!8/9/2004 Examples of Physical Quantities.doc 4/6 Jim Stiles The University of Kansas EECS Note this discrete scalar value indicates my weight at a specific time (t=0). We likewise could determine my current weight (a discrete scalar value) by evaluating the scalar field W(t) at t =41 (Doh!). 2) The current surface temperature across the entire the U.S. Again, this quantity cannot be specified with a single numeric value. Instead, we must specify temperature as a function of position (location) on the surface of the U.S. , e.g.: ( , ) 80.0 0.1 0.2 0.003 ....Txy x y xy=+−+ + where x and y are Cartesian coordinates that specify a point in the U.S. Often, we find it useful to plot this function: y x8/9/2004 Examples of Physical Quantities.doc 5/6 Jim Stiles The University of Kansas EECS Again, we can use this scalar field to determine discrete scalar values—we must simply indicate a specific location (point) in the U.S. For example: (, , ) 72 F(, , ) 97 F(, , ) 88 FTx y Seattle WATx y Dallas TXTxy ChicagoIL======DDD 88 97 72 D. Vector Fields are vector quantities that must be described as a function of (typically) space and/or time. Note that this means both the magnitude and direction of vector quantity are a function of time and/or space!8/9/2004 Examples of Physical Quantities.doc 6/6 Jim Stiles The University of Kansas EECS An example of a vector field is the surface wind velocity across the entire U.S. Again, it is obvious that we cannot express this as a discrete vector quantity, as both the magnitude and direction of the surface wind will vary as a function of location (x,y): We can mathematically describe vector fields using vector algebraic notation. For example, the wind velocity across the US might be described as: 22ˆˆ(,) (2 )xyxy xya x y a=+−v Don’t worry! You will learn what this vector field expression means in the coming weeks.8/9/2004 Vector Representations.doc 1/3 Jim Stiles The Univ. of Kansas Dept. of EECS Vector Representations We can symbolically represent a discrete vector quantity as an arrow: * The length of the arrow is proportional to the magnitude of the vector quantity. * The orientation of the arrow indicates the direction of the vector quantity. For example, these arrows symbolize vector quantities with equal direction but different magnitudes: while these arrows represent vector quantities with equal magnitudes but different directions:8/9/2004 Vector Representations.doc 2/3 Jim Stiles The Univ. of Kansas Dept. of EECS * Two vectors are equal only if both their magnitudes and directions are identical. ≠≠= * The variable names of a vector quantity will always be either boldface (e.g., AE ) or have an overbar (e.g., ,,HA, B, C ). HE We will learn that vector quantities have their own special algebra and calculus! This is why we must clearly identify vectors quantities in our mathematics (with boldface or overbars). By contrast, variables of scalar quantities will not be in bold face or have an overbar (e.g. I, V, x, ,ρφ)8/9/2004 Vector Representations.doc 3/3 Jim Stiles The Univ. of Kansas Dept. of EECS Vector algebra and vector calculus include special operations that cannot be performed on scalar quantities (and vice versa). Thus, you absolutely must denote (with an overbar) all vector quantities in the vector math you produce in homework and on exams!!! Vectors not properly denoted will be assumed scalar, and thus the mathematical result will be incorrect—and will be graded appropriately (this is bad)! * The magnitude of a vector quantity is denoted as: or EA Note that the magnitude of a vector quantity is a scalar quantity (e.g., =F 6 Newtons or =v 45