Defn: y is said to be proportional to x if there is k>0 such that y = k x.

Notation: y ~x

Note: We can have many other relationships between y and x such as y~x², y~ln(x), y~e^x. Also, y ~x implies x ~y, since k is nonzero. Similarly for the other relationships.
 

Result: Proportionality is an equivalence relation.

Pf: (Reflexive) Since x = x, then x ~x.
(Symmetric) If x ~y, then x = k y or y = (1/k) x, or y ~x.
(Transitive) If x ~y and y ~z, then x ~z is easy by combining terms.

Famous Proportionalities: - page 100

• Hooke's Law: F = k x
• Newton's Law: F = m a
• Ohm's Law: V = i R
• Boyle's Law: V = k/P (inversely proportional)
• Relativity: E = c² M
• Kepler's Third Law: T = c R^3/2

Defn: Two objects are said to be geometrically similar if there is a 1-1 correspondence between points of the objects such that the ratio of distances between corresponding points is constant for all possible pairs of points.

(Objects must be scaled replicas of each other.)
a
Projects are in the remaining sections of chapter four.

References:

Introduction to Applied Mathematics, Gilbert Strang, Wellesley Cambridge Press, 1986

Applied Linear Algebra, Ben Noble and James Daniel, Prentice-Hall, 1977
 
 

Approaches to use when viewing data:

• Model Fitting (Approximation) - starting with the data, create a formula which "closely"correlates
• Interpolation - starting with a general formula, determine parameters to force it to fit the data exactly.

• Fitting a selected model type or types to the data
• Choosing the most appropriate model from competing types that have been fitted.

• Formulation - using incorrect assumptions or leaving out important facts
• Truncation - taking an exact process and approximating it
• Convergence - taking a convergent process and stopping it after a finite number of steps
• Round-off - computer arithmetic is very seldom exact
• Measurement - imprecise data

• Absolute error - measures absolute deviation between exact and approximation. Generally, we only measure absolute "vertical" error in determining how well our model y = f(x) fits the given data. So, given data points (x₀,y₀), ... ,(x_n,y_n), the absolute error at the kth point will be the quantity | y_k - f(x_k) |.

• In fitting data to a model, we will expect some error since our data is most likely subject to measurement error. However, if the model is to indeed approximate the data, this error ought to be small.
• Considering a model of the form y=f(x), we can plot (f(x_k), y_k ) and check for a straight line.
• We can manipulate the for y = f(x) via some other invertible function g(x) ( such as g(x)=ln(x) ) and plot ( g(f(x_k)), g(y_k) ) and check for a straight line.

• Linear Least Squares
• Polynomial Least Squares
• Transformed Least Sqares