54  Glossary

apply the function,1,01 applying,1,01 axis,1,01 convert,1,01 coordinate plane,1,01 degree of freedom,1,01 dimension,2,01 domain,4,01 functions,3,01 inputs,1,01 maps,1,01 mathematical models,1,01 model,1,01 name of an input,1,01 not a symbolic name,1,01 number line,2,01 output,4,01 quantity,4,01 range,1,01 scale,1,01 trajectory,3,01 application of a function to an input,1,02 apply a function,1,02 argument,1,02 constants,1,02 defining,1,02 evaluate a function on,1,02 formula,3,02 notation,1,02 parameters,4,02 period,4,02 special inputs,2,02 declarative,1,03 do some action,1,03 evaluating,1,03 format,1,03 imperative,1,03 only,1,03 storage arrow,1,03 storage,1,03 tilde expression,2,03 constant value,1,04 contour plot,2,04 function,12,04 graph,3,04 graphical domain,1,04 graphs,1,04 one,1,04 output space,1,04 slice,3,04 space,1,04 surface,2,04 base,1,05 bit,1,05 decade,1,05 doubling time,1,05 gaussian,1,05 inversely proportional,1,05 monomials,2,05 natural logarithm,2,05 negative powers,1,05 non-integer powers,1,05 normal function,1,05 not,8,05 pattern-book functions,2,05 power-law functions,1,05 sigmoidal,1,05 argmax,8,06 argmin,5,06 asymptotic,1,06 at,1,06 change in slope,1,06 change,1,06 concave down,3,06 concave up,4,06 concavity,5,06 “continuous on an interval (a,b)”,1,06 continuous,4,06 cycle,1,06 decrease,1,06 divided by,1,06 extrema,1,06 horizontal asymptote,1,06 increase,1,06 inflection points,1,06 input,5,06 local maximum,3,06 local minimum,3,06 monotonic,2,06 negative concavity,1,06 not include 0,1,06 of,1,06 optimization,2,06 periodic,1,06 positive concavity,1,06 rate of change,3,06 vertical asymptote,2,06 continuous domain,1,07 data frame,3,07 discrete,1,07 evaluate,2,07 explanatory variables,2,07 interpolation,4,07 machine learning,1,07 point plot,1,07 response,1,07 statistics,1,07 variable,1,07 black,4,08 input scaling,2,08 output scaling,2,08 parameter,2,08 pure number,1,08 pure numbers,1,08 quantities,2,08 coefficients,2,09 constant of proportionality,1,09 discontinuity,2,09 exterior function,2,09 fitting the function,1,09 function composition,1,09 inner,1,09 interior function,2,09 linear combination,4,09 local behavior,1,09 outer,1,09 piecewise function,1,09 wave packet,1,09 without lifting the pencil from the paper,1,09 displacement,3,10 just the identity function,1,10 multiplication,1,10 10,1,11 amplitude,1,11 baseline,1,11 fitting,1,11 half,1,11 layers,1,11 modeling cycle,4,11 newton’s law of cooling,2,11 polishing,1,11 regression,1,11 constant term,2,12 demand functions,2,12 eight simple shapes,6,12 interaction term,3,12 linear terms,1,12 order,1,12 polynomial,1,12 production functions,2,12 quadratic terms,2,12 information,1,13 inverting,1,13 iterate,4,13 iteration,1,13 maximization,1,13 minimization,1,13 zero finding,2,13 decades,1,14 difference,1,14 log-log,1,14 logarithm,1,14 lux,1,14 magnitude,2,14 negative values,1,14 order of magnitude,1,14 orders of magnitude,1,14 ratio,1,14 scientific notation,1,14 semi-log,1,14 units,1,14 [acceleration],1,15 [area],1,15 [energy],1,15 [force],1,15 [momentum],1,15 [power],1,15 [velocity],1,15 [volume],1,15 coefficients must themselves have dimension,1,15 compound dimensions,1,15 degrees,1,15 density,2,15 dimensionless,1,15 divide,3,15 flavors of one,1,15 fundamental dimensions,1,15 gravity,2,15 length,4,15 mass,3,15 multiply,2,15 never,1,15 radians,2,15 time,3,15 unit,2,15 work,2,15 data,1,16 empirical models,1,16 extrapolation,1,16 hypothetico-deductive model,1,16 mechanistic models,1,16 observation,1,16 residuals,1,16 scientific method,1,16 systematic,1,16 54,1,17 differentiation,4,17 elliptical,1,17 flow,2,17 water volume,1,17 with-respect-to,1,17 average rate of change,1,18 instantaneous slope function,1,18 instantaneous slope,1,18 model the model,1,18 objective function,4,18 rate,1,18 slope function,1,18 velocity,2,18 non-standard analysis,1,19 derivative,2,19 theory of limits,1,19 finite-difference,1,20 second derivative,4,20 static,1,20 symbolic differentiation,1,20 change in the slope,1,21 constant,1,21 curvature,3,21 first derivative,1,21 radius of curvature,1,21 smooth,1,21 smoothness,3,21 third derivative,2,21 transition,2,21 twice,1,21 you are not expected to memorize it,1,21 c-infinity,1,22 continuity,1,22 continuous sets,1,22 cubic splines,1,22 differencing operator,1,22 discrete sets,1,22 example of non-\(c^2\) functions:,1,22 examples of \(c^\infty\) functions,1,22 heaviside function,1,22 ramp function,1,22 real numbers,2,22 small change in input,1,22 small change in output,1,22 adding,1,23 chain rule,1,23 composition of two functions,1,23 composition,1,23 derive,1,23 h-theory,1,23 left graph,1,23 linear operator,1,23 per capita,1,23 per,8,23 percent,1,23 power-law rule,1,23 product of two functions,1,23 product rule,2,23 proportional increase,1,23 right graph,1,23 rules,1,23 scaling,2,23 step 1,1,23 step 2,1,23 step 3,1,23 arg,1,24 critical points,1,24 decision quantity,1,24 differential equations,3,24 max,2,24 maximize,1,24 maximum,2,24 micron,1,24 minimize,1,24 minimum,1,24 multiple objectives,1,24 negative,2,24 objectives,1,24 only part,1,24 “out of 55,000”,1,24 parabola,1,24 positive,1,24 precision,1,24 revenue,1,24 slope,4,24 value of the second derivative,1,24 value,1,24 all other things being equal,1,25 all other things held constant,1,25 blue,1,25 facet,1,25 gradient field,1,25 gradient vector,2,25 gradient,2,25 green,1,25 held constant,1,25 orientation,1,25 partial derivative,1,25 partial derivatives,1,25 sum,1,25 tangent plane,1,25 linear regression,1,26 linear,1,26 local,2,26 mixed partial derivatives,1,26 quadratic,1,26 second-order polynomial,1,26 error,2,27 exact,1,27 far,1,27 indeterminant form,1,27 l’hopital’s rule,1,27 least squares,1,27 n-th order,1,27 near,3,27 tail domains,1,27 taylor polynomial,2,27 well behaved,1,27 wriggly domain,1,27 ** componentwise ,1,28 alignment,1,28 angle,2,28 column vector,1,28 column,1,28 correlation coefficient,1,28 cosine of the angle,2,28 degrees of freedom,1,28 direction,2,28 dot product,3,28 embedded,1,28 functional notation,1,28 functional,1,28 infix notation,1,28 linear algebra,1,28 matrix mechanics,1,28 mean,3,28 negative correlation,1,28 orthogonal,2,28 positively correlated,1,28 protractor,1,28 row vector,1,28 ruler,1,28 scalar,2,28 scale a vector,1,28 square length,1,28 step,2,28 subspace,4,28 tail,1,28 tip,1,28 variance,6,28 vector space,1,28 vectors,3,28 zero correlation,1,28 bold-faced,1,29 componentwise multiplication,1,29 matrix multiplication,1,29 matrix,2,29 projected onto,1,29 scaled,1,29 subspace spanned,1,29 subspaces,1,29 subtraction,1,29 table,1,29 tail to tail,1,29 vector,1,29 at the same time,1,30 decomposing,1,30 decomposition,1,30 model space,1,30 model vector,1,30 mutually orthogonal,2,30 not mutually orthogonal,1,30 projecting,1,30 projection,1,30 residual vector,1,30 target vector,1,30 unit length vector,1,30 both,4,31 columns of data,1,31 projection problem,2,31 target problem,4,31 accounted for,1,32 analysis of variance,1,32 artificial intelligence,1,32 body mass index,1,32 genetics,1,32 intercept,1,32 maximum likelihood estimation,2,32 r-squared,1,32 response variable,1,32 unaccounted for,1,32 unexplained,1,32 variation,1,32 anti-differentiation,2,33 brackets,1,33 with-respect-to input,1,33 age of the universe,1,35 anti-derivative,1,35 apply,2,35 bold-face,3,35 bounds of integration,1,35 constant of integration,3,35 definite integral,3,35 dummy variable,1,35 first fundamental theorem of calculus,1,35 flavor of one,1,35 force accumulated over time,1,35 force,2,35 indefinite integral,1,35 lower bound of integration,1,35 momentum,2,35 net change,2,35 second fundamental theorem of calculus,1,35 upper bound of integration,1,35 analyze,1,36 decompose,1,36 fourier transform,2,36 frequency domain,1,36 harmonics,1,36 knots,2,36 molecular vibrational spectroscopy,1,36 periodicity,1,36 time domain,1,36 waveforms,1,36 cumulative sum,1,37 cumulative,1,37 dynamics,3,37 euler steps,1,37 gauss-legendre quadrature,1,37 integrand,1,37 series of quantities,1,37 cataloged functions,2,38 elementary functions,1,38 helper function,1,38 no such rules,1,38 normal pdf,1,38 parts,2,38 slope function visualization,1,38 symbolic anti-derivatives,1,38 symbolic derivatives,1,38 techniques of integration,1,38 u-substitution,1,38 a trajectory can never cross itself,1,39 change in state,1,39 chaos,2,39 “continuous-time, continuous-space system”,1,39 deterministic,2,39 differential equation,1,39 dimension of the state space,1,39 “discrete-time, discrete-state dynamical system”,1,39 dynamical functions,1,39 dynamical state,1,39 dynamical system,1,39 dynamical systems,1,39 essential,1,39 first-order differential equations,2,39 fixed point,3,39 function of time,1,39 gradient ascent,2,39 initial condition,3,39 instantaneous state,2,39 oscillates,1,39 path,1,39 position,2,39 second-order differential equation,3,39 solution,1,39 speed,2,39 state space,4,39 state velocity,3,39 state,1,39 stochastic,2,39 system of differential equations,1,39 system,1,39 time series,1,39 accumulating,1,40 ansatz,2,40 continuous-time function,1,40 euler formula,1,40 euler method,1,40 graphical method,1,40 integrating,1,40 limit definition of the derivative,1,40 logistic equation,1,40 method of ansätze,1,40 solutions,1,40 substitution methods,1,40 catastrophe theory,1,41 tipping point,1,41 evaluated,1,41 fixed points,2,41 flow field,1,41 linear approximation,1,41 linearization,1,41 net,1,41 stability,1,41 stable,2,41 sustainability,1,41 unstable,1,41 neutral stability,1,42 nullclines,1,42 saddle,1,42 saddles,1,42 sink,1,42 source,1,42 vector field,1,42 bounded stable model,1,43 bounded unstable model,1,43 carrying capacity,1,43 chain reaction,1,43 conserved,2,43 coupled,1,43 critical mass,1,43 explosion,1,43 fissile,1,43 framework,1,43 function shape,1,43 interaction,1,43 linear stable model,1,43 linear unstable model,1,43 love,1,43 near zero,1,43 phase plane,1,43 predator-prey model,1,43 radioactive decay,1,43 reactor,1,43 sir model,1,43 square of the army size,1,43 the same problem,1,43 twice as capable,1,43 two,1,43 warfare,1,43 “ in the language of dynamical systems, the equilibrium state of a system is called a “,1,44 equilibrium,2,44 steady state,1,44 transient,2,44 cruise control,1,45 eigenvalues,2,45 eigenvectors,1,45 angular frequency,1,46 ballistics,1,46 complex number,1,46 critically damped,1,46 damping,1,46 force-balance equations,1,46 fundamental theorem of calculus,1,46 law ii,1,46 must have,1,46 theory of general relativity,1,46 accumulate,1,47 anti-differentiate the function,1,47 argmax finding,1,47 computational mathematicians,1,47 differentiate the function,1,47 equations,1,47 evaluating a function,1,47 ideal result,1,47 integrate the function,1,47 logistic map,1,47 mathematical modeling,1,47 mathematical task,1,47 name,1,47 newton step,1,47 number,1,47 operations on functions,1,47 realistic result,1,47 solving,1,47 zero-finding,1,47 ad hoc,1,48 control point,1,48 cubic segments,1,48 cubic spline,2,48 five,1,48 function fitting,1,48 interpolating function,1,48 mandate,1,48 s,2,48 single valued,1,48 six,1,48 stiff,1,48 stiffness,1,48 aids to human decision making,1,49 budget constraint,1,49 cobb-douglas,1,49 constraint,1,49 dominated,1,49 dynamic programming,1,49 equality constraint,1,49 equality constraints,1,49 evaluation phase,1,49 gradient descent,1,49 incommensurate,1,49 inequality constraint,1,49 iterative,1,49 lagrange multiplier,2,49 linear programming,1,49 min,1,49 modeling phase,1,49 operations research,1,49 parallel,1,49 potential energy,1,49 quadratic programming,1,49 relaxed,1,49 shadow price,1,49 simplex,1,49 solution phase,1,49 spring-mass systems: an example context,1,49 sub-optimal,1,49 vector input,1,49 arithmetic mean,1,50 bayes’ rule,1,50 bayesian inference,1,50 center of mass,1,50 event,1,50 expected value,2,50 exponential probability density,1,50 gaussian density function,1,50 hypothesis,1,50 hypothetical reasoning,1,50 likelihood function,1,50 likelihood functions,1,50 normal distribution,1,50 normalize,1,50 normalizing,1,50 not a probability,1,50 not definitive,1,50 outcome,1,50 posterior functions,1,50 posterior,1,50 prior functions,1,50 prior,1,50 priors,1,50 probability density function,1,50 probability density functions,2,50 probability density,3,50 probability,1,50 relative density,1,50 relative density function,1,50 relative density functions,1,50 risk of disease transmission,1,50 sd,1,50 standard deviation,2,50 uncertainty,1,50 under construction,1,50 uniform density function,1,50 update,1,50 compound interest,1,51 constrained optimization,1,51 discount,1,51 discounting function,1,51 discounting,1,51 interest rate,1,51 interest rates,1,51 interest,1,51 mortgage,1,51 net present value,1,51 nominal income stream,1,51 present value,3,51 stream,1,51 term,1,51 transitive,1,51 votes,1,51 more precisely it can be called ,1,52 angular acceleration,1,52 area,1,52 conservation of momentum,1,52 decomposed,1,52 decomposing a vector,1,52 energy,1,52 finite,1,52 forces,1,52 infinitesimal,1,52 instantaneous length,1,52 instantaneous rate of change,1,52 kinetic energy,1,52 latent,1,52 mechanics,1,52 rotation,1,52 statics,1,52 study of machines,1,52 torque,1,52 turbofan engines,1,52 turbojet engines,1,52 diffusion equation,1,53 essay,1,53 heat equation,1,53 optional background,1,53 quick review questions,1,53 random walk,1,53 three,1,53 wrongly**,1,53