Applications, Examples and Libraries

Photon Exposure

by: John Dolese 125 Product: Maple 18

September 26 2014

5 0

This application calculates the number of photons reaching a camera sensor for a given exposure. A blackbody model of the sun is generated. The "Sunny 16" rule for exposure is demonstrated. Calculations are done using units.Photon_Exposure_Array.mw

Photon Exposure

Blackbody Model of the Sun

	Plank Constant
	Boltzman Constant
	Light Speed
	Sun Radius
	Earth Orbit
	Sun Color Temperature
	Transmission Factor

Sun: Spectral Radiant Exitance to Earth: Spectral Irradiance

"M(lambda):=(2*Pi*h*c^(2))/((lambda)^(5))*1/((e)^((h*c)/(lambda*kb*Tsun))-1)*(Rsun/(Re_orb))^(2)*tf_atm:"

=

Photopic Relative Response VP vs λ

=

= $Vector(4, {(1) = ` 471 x 2 `*Matrix, (2) = `Data Type: `*float[8], (3) = `Storage: `*rectangular, (4) = `Order: `*Fortran_order})$

(ArrayInterpolation for x,y data VPdata returns y' for new x data lambdaP)

Illuminance in Radiometric and Photometric Units:

E__r := sum(Units:-Standard:-`*`(M(Units:-Standard:-`*`(lambda, Unit('nm'))), Unit('nm')), lambda = 200 .. 4000) =

E__po := Units:-Standard:-`*`(Units:-Standard:-`*`(683.002, Units:-Standard:-`*`(Unit('lm'), Units:-Standard:-`/`(Unit('W')))), sum(Units:-Standard:-`*`(Units:-Standard:-`*`(VP[lambda], M(Units:-Standard:-`*`(lambda, Unit('nm')))), Unit('nm')), lambda = 200 .. 4000)) =

Translation from Illuminance to Luminance for Reflected Light;

Object Reflectance

Object Luminance L__po := proc (R__o) options operator, arrow; R__o*E__po/(Pi*Unit('sr')) end proc: =

Illuminance of a Camera Sensor E_ps applied for time t_expdetermines Luminous Exposure H_p;

Ideal Illuminance is determined by the exposure time t_exp, effective f-number N and to a less extent the angle to the optical axis θ;

•	H Luminous Exposure

•	E_ps Illuminance to the Camera

•	N Effective F-Number

•	t_exp Exposure Time

•	θ Angle to the Optical Axis

E__ps_ideal = Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`/`(4)), L__po), Units:-Standard:-`*`(Units:-Standard:-`^`(cos(theta), 4), Units:-Standard:-`/`(Units:-Standard:-`^`(N, 2)))):

The camera meter determines the exposure time t_expto balance the object luminance, reflectance and effective f-number. It does this based on an internal constant k and the camera ISO s.

•	s ISO Gain (Based on saturation at 3 stops above the average scene luminance)

•	k Reflected Light Meter Calibration Constant

for Nikon, Canon and Sekonic

•	c Incident Light Meter Calibration Constant

for Sekonic with flat dome

$N^2/t__exp = `#mrow(mi("\`E__po\`"),mo("⋅"),mi("s"))`/c__m$ (Incident Light Meter)

$Units:-Standard:-`*`(Units:-Standard:-`^`(N, 2), Units:-Standard:-`/`(t__exp)) = Units:-Standard:-`*`(`#mrow(mi("\`L__po\`"),mo("⋅"),mi("s"))`, Units:-Standard:-`/`(k__m)):$ (Reflected Light Meter)

Solve for H in terms of the Camera Meter Constant k and s

Es = Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`/`(4)), Lo), Units:-Standard:-`*`(Units:-Standard:-`^`(cos(theta), 4), Units:-Standard:-`/`(Units:-Standard:-`^`(N, 2)))):

H = Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`/`(4)), Lo), Units:-Standard:-`*`(Units:-Standard:-`^`(cos(theta), 4), Units:-Standard:-`/`(Units:-Standard:-`^`(N, 2)))), Units:-Standard:-`*`(Units:-Standard:-`*`(km, Units:-Standard:-`^`(N, 2)), Units:-Standard:-`/`(Units:-Standard:-`*`(Lo, s)))) H = (1/4)*Pi*cos(theta)^4*km/s

H__p := proc (s, theta) options operator, arrow; (1/4)*Pi*k__m*cos(theta)^4/s end proc:

=

Note: Meters are typically set for a scene reflectance 3 stops below 100% or 12.5%.

E__ps := proc (N, R__o, theta) options operator, arrow; (1/4)*Pi*Unit('sr')*R__o*E__po*cos(theta)^4/(Pi*Unit('sr')*N^2) end proc:

=

t__exp_ideal := proc (N, s, R__o) options operator, arrow; H__p(s, theta)/E__ps(N, R__o, theta) end proc:

=

Actual exposure time includes typical lens losses;

	Magnification
	Lens Transmittance
	Lens Flare
	Vignetting

Total Lens Efficiency

=

Replacing Eps with q*Eps we get the "Sunny 16" relation between exposure time and ISO;

t__exp := proc (N, s, R__o) options operator, arrow; H__p(s, theta)/(q*E__ps(N, R__o, theta)) end proc: =

t__exp_alt := proc (N, s, R__o) options operator, arrow; k__m*N^2*Pi/(s*q*R__o*E__po) end proc: =

•	The Number of Photons NP Reaching the Sensor Area A;

•	Circle of confusion for 24x36mm "Full Frame" for 1 arcminute view at twice the diagonal:

A__cc := Units:-Standard:-`*`(Units:-Standard:-`*`(Pi, Units:-Standard:-`^`(Units:-Standard:-`*`(12.6, Unit('`μm`')), 2)), Units:-Standard:-`/`(4)):

•	Sensor Bandwidth Photopic Response VP

•	Exposure Time for Zone 5: Rscene=12.5% , Saturation in Zone 8 Rscene=100%

•	Camera ISO differs from Saturation ISO. Typical Saturation ISO is 2300 when the camera is set to 3200. See DxoMark.

The average number of photons for exposure time based on Reflectance of the scene relative to the metered value:

Zone 5;

NP := proc (s, R__o, theta) options operator, arrow; (1/4)*t__exp(N, s, R__meter)*A__cc*q*R__scene*cos(theta)^4*(sum(VP[lambda]*M(lambda*Unit('nm'))*Unit('nm')*lambda*Unit('nm')/(h*c), lambda = 200 .. 4000))/N^2 end proc:

=

Zone 8; R__meter := Units:-Standard:-`*`(R__scene, Units:-Standard:-`/`(Units:-Standard:-`^`(2, 3))):

NP__sat := proc (s, theta) options operator, arrow; (1/4)*t__exp(N, s, R__meter)*A__cc*q*R__scene*cos(theta)^4*(sum(VP[lambda]*M(lambda*Unit('nm'))*Unit('nm')*lambda*Unit('nm')/(h*c), lambda = 200 .. 4000))/N^2 end proc:

=

Approximate Formula

H__sat := proc (s__sat) options operator, arrow; H__p(s__sat, 0)*E__ps(N, 1, 0)/E__ps(N, 1/8, 0) end proc :

= HFloat(78.53981635)*Units:-Unit(('cd')*('s')/('m')('radius')^2)/s__sat

Average Visible Photon Energy

P__e_ave := Units:-Standard:-`*`(Units:-Standard:-`/`(Units:-Standard:-`+`(850, -350)), sum(Units:-Standard:-`*`(Units:-Standard:-`*`(h, c), Units:-Standard:-`/`(Units:-Standard:-`*`(lambda, Unit('nm')))), lambda = 350 .. 850)): =

NPtyp := proc (s__sat) options operator, arrow; H__sat(s__sat)*A__cc/(683.002*(Unit('lm')/Unit('W'))*P__e_ave) end proc:

=

Download Photon_Exposure_Array.mw

Create DNG Matrices Using Optimization

by: John Dolese 125 Product: Maple 18

September 20 2014

2 0

Obsolete

See my Camera Profiler application instead.

This application creates DNG matrices by optimizing Delta E from a raw photo of x-rites color checker. The color temperature for the photograph is also estimated. Inputs are raw data from RawDigger and generic camera color response from DXO Mark.

	Initialization

XYZoptical to RGB to XYZdata

Sr,g,b is the relative spectral transmittance of the filter array not selectivity for XY or Z of a given color.

Pulling Sr,g,b out of the integral assumes they are scalars. For example Sr attenuates X, Y and Z by the same amount.

Raw Balance is not White Point Adaptation.

The transmission loss of Red and Blue pixels relative to green is compensated by D=inverse(S). The relation to incident chromaticity, xy is unchanged as S.D=1.

(See Bruce Lindbloom; "Spectrum to XYZ" and "RGB/XYZ Matrices" also, Marcel Patek; "Transformation of RGB Primaries")

•	XYZ to RGB

$(Vector(3, {(1) = R_Tbb, (2) = G_Tbb, (3) = B_Tbb})) = (Matrix(3, 3, {(1, 1) = XR*Sr, (1, 2) = YR*Sr, (1, 3) = ZR*Sr, (2, 1) = XG*Sg, (2, 2) = YG*Sg, (2, 3) = ZG*Sg, (3, 1) = XB*Sb, (3, 2) = YB*Sb, (3, 3) = ZB*Sb})).(Vector(3, {(1) = X_Tbb, (2) = Y_Tbb, (3) = Z_Tbb}))$

$(Vector(3, {(1) = R_Tbb, (2) = G_Tbb, (3) = B_Tbb})) = (Matrix(3, 3, {(1, 1) = Sr, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = Sg, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = Sb})).(Matrix(3, 3, {(1, 1) = XR, (1, 2) = YR, (1, 3) = ZR, (2, 1) = XG, (2, 2) = YG, (2, 3) = ZG, (3, 1) = XB, (3, 2) = YB, (3, 3) = ZB})).(Vector(3, {(1) = X_Tbb, (2) = Y_Tbb, (3) = Z_Tbb}))$

$Camera_Neutral = (Matrix(3, 3, {(1, 1) = Sr, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = Sg, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = Sb})).(Matrix(3, 3, {(1, 1) = XR, (1, 2) = YR, (1, 3) = ZR, (2, 1) = XG, (2, 2) = YG, (2, 3) = ZG, (3, 1) = XB, (3, 2) = YB, (3, 3) = ZB})).(Vector(3, {(1) = X_wht, (2) = Y_wht, (3) = Z_wht}))$

•	RGB to XYZ (The extra step of adaptation to D50 is included below)

$(Vector(3, {(1) = X_D50, (2) = Y_D50, (3) = Z_D50})) = (Matrix(3, 3, {(1, 1) = XTbbtoXD50, (1, 2) = YTbbtoXD50, (1, 3) = ZTbbtoXD50, (2, 1) = XTbbtoYD50, (2, 2) = YTbbtoYD50, (2, 3) = ZTbbtoYD50, (3, 1) = XTbbtoZD50, (3, 2) = YTbbtoZD50, (3, 3) = ZTbbtoZD50})).(Matrix(3, 3, {(1, 1) = RX*Dr, (1, 2) = GX*Dg, (1, 3) = BX*Db, (2, 1) = RY*Dr, (2, 2) = GY*Dg, (2, 3) = BY*Db, (3, 1) = RZ*Dr, (3, 2) = GZ*Dg, (3, 3) = BZ*Db})).(Vector(3, {(1) = R_Tbb, (2) = G_Tbb, (3) = B_Tbb}))$

$(Vector(3, {(1) = X_D50, (2) = Y_D50, (3) = Z_D50})) = (Matrix(3, 3, {(1, 1) = XTbbtoXD50, (1, 2) = YTbbtoXD50, (1, 3) = ZTbbtoXD50, (2, 1) = XTbbtoYD50, (2, 2) = YTbbtoYD50, (2, 3) = ZTbbtoYD50, (3, 1) = XTbbtoZD50, (3, 2) = YTbbtoZD50, (3, 3) = ZTbbtoZD50})).(Matrix(3, 3, {(1, 1) = RX, (1, 2) = GX, (1, 3) = BX, (2, 1) = RY, (2, 2) = GY, (2, 3) = BY, (3, 1) = RZ, (3, 2) = GZ, (3, 3) = BZ})).(Matrix(3, 3, {(1, 1) = Dr, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = Dg, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = Db})).(Vector(3, {(1) = R_Tbb, (2) = G_Tbb, (3) = B_Tbb}))$

$(Vector(3, {(1) = X_D50, (2) = Y_D50, (3) = Z_D50})) = (Matrix(3, 3, {(1, 1) = RX_D50, (1, 2) = GX_D50, (1, 3) = BX_D50, (2, 1) = RY_D50, (2, 2) = GY_D50, (2, 3) = BY_D50, (3, 1) = RZ_D50, (3, 2) = GZ_D50, (3, 3) = BZ_D50})).(Matrix(3, 3, {(1, 1) = Dr, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = Dg, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = Db})).(Vector(3, {(1) = R_Tbb, (2) = G_Tbb, (3) = B_Tbb}))$

$(Vector(3, {(1) = X_D50wht, (2) = Y_D50wht, (3) = Z_D50wht})) = (Matrix(3, 3, {(1, 1) = RX_D50, (1, 2) = GX_D50, (1, 3) = BX_D50, (2, 1) = RY_D50, (2, 2) = GY_D50, (2, 3) = BY_D50, (3, 1) = RZ_D50, (3, 2) = GZ_D50, (3, 3) = BZ_D50})).(Matrix(3, 3, {(1, 1) = Dr, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = Dg, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = Db})).Camera_Neutral$

	Functions

	Input Data

	Solve for Camera to XYZ D50 and T

Special bounded partial sums

by: Marvin Ray Burns 550

September 10 2014

0 1

For all real a, the partial sums s_n= sum((-1)^k (k^(1/k) -a), k=1..n) are bounded so that their limit points form an interval [-1.+ the MRB constant +a, MRB constant] of length 1-a, where the MRB constant is limit( $sum((-1)^k*(k^(1/k)), k = 1 ..2*N)$ ,N=infinity).

For all complex z, the upper limit point of s_n= sum((-1)^k (k^(1/k) -z), k=1..n) is the the MRB constant.

We see that maple knows the basics of this because when we enter sum((-1)^k*(k^(1/k)-z), k = 1 .. n)

maple gives

$sum((-1)^k*(k^(1/k)-z), k = 1 .. n)$ .

marvinrayburns.com

Banque de questions MapleTA en français – plus...

by: jzivku 640

September 05 2014

1 0

Aujourd’hui, je suis ravis d’annoncer la disponibilité d’une large banque de questions françaises supportant les enseignements du secondaire et de l’enseignement supérieur. Ce contenu didactique est disponible via le MapleTA Cloud, et également grâce au lien de téléchargement ci-dessous.

Lien de téléchargement de la banque de questions françaises

Ces questions sont librement et gratuitement accessibles, et vous pouvez les utiliser directement sur vos propres évaluations et exercices dans MapleTA, ou les éditer et modifier pour les adapter à vos besoins.

Le contenu de cette banque de questions françaises traite de sujets pour les classes et enseignements pré-bac, post-bac pour en majorité les matières scientifiques.

Les matières traitées par niveaux et domaines sont:

Lycées :

Electricité
Équations Différentielles
Gravitation universelle
Langues
Maths I
Maths II
Physique
Chimie
Mécanique

Enseignement supérieur (Post-Bac) :

Astrobiologie
Introduction au Calcul pour la Biologie
Chimie
Déplacement d'onde
Electricité & Magnétisme
Maths pour l’économie
Maths Post-Bac
Mécanique Angulaire
Mécanique des Fluides
Mécanique linéaire
Physique Post-Bac
Electrocinétique
Matériau
Mécanique des Fluides
Thermodynamique

Jonny Zivku
Maplesoft Product Manager, Maple T.A.

Maple T.A. course materials – Introductory Calculus ...

by: jzivku 640

August 27 2014

1 0

Several Maple T.A. users have developed comprehensive sets of question content and assignments to support full courses in Maple T.A. These questions are available through the Maple T.A. Cloud, and we have decided to also post the associated course modules on Maple Primes as an alternative way of accessing this content.

Below you will find a link to the Introductory Calculus Maple T.A.. course module developed by Keele University.

This testing content is freely distributed, and can be used in your own Maple T.A. tests either as-is, or with edits.

These questions are designed to accompany the first semester of an introductory honours calculus course. The course is intended primarily for students who need or expect to pursue further studies in mathematics, physics, chemistry, engineering and computer science. With over 250 question, topics include: basic material about functions, polynomials, logs and exponentials, the concept of the derivative, and lots of practise exercises for finding derivatives and integrals, and material about series.

Jonny Zivku
Maplesoft Product Manager, Maple T.A.

Maple T.A. course materials – Introductory Calculus...

by: jzivku 640

August 27 2014

3 4

Several Maple T.A. users have developed comprehensive sets of question content and assignments to support full courses in Maple T.A. These questions are available through the Maple T.A. Cloud, and we have decided to also post the associated course modules on Maple Primes as an alternative way of accessing this content.

Below you will find a link to the Introductory Calculus for Biological Sciences Maple T.A.. course module developed by the University of Guelph.

This testing content is freely distributed, and can be used in your own Maple T.A. tests either as-is, or with edits.

The Introductory Calculus for Biological Sciences course module is designed to cover a single-semester introductory calculus course for biological sciences students at the first-year university level. The questions are designed to span the topics listed below, allowing for practice, homework or testing throughout the semester.

Topics include:

Introduction to Functions
Composite and Inverse Functions
Trigonometric Functions
Logarithms and Exponents
Sequences and Finite Series
Limits and Continuity
Derivatives
Curve Sketching
Differentials
Linear Approximation
Taylor Polynomials
Difference Equations
Log-Log Graphs
Anti-Differentiation
Definite Integrals

Jonny Zivku
Maplesoft Product Manager, Maple T.A.

Maple T.A. course materials – Introductory Mathema...

by: jzivku 640

August 22 2014

0 0

Several Maple T.A. users have developed comprehensive sets of question content and assignments to support full courses in Maple T.A. These questions are available through the Maple T.A. Cloud, and we have decided to also post the associated course modules on Maple Primes as an alternative way of accessing this content.

Below you will find a link to the Introductory Mathematical Economics Maple T.A.. course module developed by the University of Guelph.

This testing content is freely distributed, and can be used in your own Maple T.A. tests either as-is, or with edits.

The Introductory Mathematical Economics course module is designed to cover a single-semester course in mathematical economics for economics and commerce students at the second-year university level. The questions are designed to span the topics listed below, allowing for practice, homework or testing throughout the semester.

Topics include:

Rules of Differentiation
First Order Differential Equations
Higher Order Derivatives
Optimization in One Variable
Second Order Conditions for Optimization
Systems of Linear Equations
Optimization with Direct Restrictions on Variables
Over Determined and Under Determined Systems
Matrix Representation of Systems
Gauss Jordan
Matrix Operations
Types of Matrices
Determinants and Inverses
Partial Differentiation
Second Order Partial Derivatives
Multivariate Optimization
Second Order Conditions for Multivariate Optimization
Multivariate Optimization with Direct Restrictions of Variables
Constrained Optimization and the Lagrangean Method
Second Order Conditions for Constrained Optimization

Jonny Zivku
Maplesoft Product Manager, Maple T.A.

Maple T.A. course materials – Introductory Electricity...

by: jzivku 640

August 22 2014

1 0

Several Maple T.A. users have developed comprehensive sets of question content and assignments to support full courses in Maple T.A. These questions are available through the Maple T.A. Cloud, and we have decided to also post the associated course modules on Maple Primes as an alternative way of accessing this content.

Below you will find a link to the Introductory Electricity & Magnetism Maple T.A.. course module developed by the University of Guelph.

This testing content is freely distributed, and can be used in your own Maple T.A. tests either as-is, or with edits.

The Introductory Electricity & Magnetism course module is designed to cover a single-semester course in electricity and magnetism for physical sciences students at the first-year university level. The questions are designed to span the topics listed below, allowing for practice, homework or testing throughout the semester. Using the Maple engine that is part of Maple TA, a custom grading engine has been developed to provide even more flexible grading of scalar and vector responses. This partial grading engine can be configured to, among other things, assign part marks for missing units, transposed or missing vector components or missing algebraic terms.

Topics include:

Cross Products
Coulomb’s Law
Electric Fields
Point Charge Distributions
Continuous Charge Distributions (Integration)
Electric Potential
Electric Potential Energy
Electromotive Force
Resistance
Capacitance
Kirchhoff’s Laws
Magnetic Fields
Magnetic Fields Due to Current Carrying Wires
Forces on Wires in Magnetic Fields
Forces on Charges in Electric and/or Magnetic Fields
EM Waves
Two Source Interference
Double Slit Interference
Single Slit Diffraction
Diffraction Gratings

Jonny Zivku
Maplesoft Product Manager, Maple T.A.

Maple T.A. course materials – Statistics (University...

by: jzivku 640

August 20 2014

1 0

Several Maple T.A. users have developed comprehensive sets of question content and assignments to support full courses in Maple T.A. These questions are available through the Maple T.A. Cloud, and we have decided to also post the associated course modules on Maple Primes as an alternative way of accessing this content.

Below you will find a link to the Statistics Maple T.A.. course module developed by the University of Guelph.

This testing content is freely distributed, and can be used in your own Maple T.A. tests either as-is, or with edits.

The Statistics course module is designed to cover a single-semester course in statistics for science students at the second-year university level. The questions are designed to span the topics listed below, allowing for practice, homework or testing throughout the semester. The questions are mainly of an applied nature and do not delve very deeply into the underlying mathematical theory.

Topics:

Introduction to Statistics
Descriptive Statistics
Basic Probability
Discrete Random Variables
Continuous Random Variables
Sampling Distributions
Inference for Means
Inference for Proportions
Inference for Variances
Chi-square Tests for Count Data
One-Way ANOVA
Simple Linear Regression and Correlation

Jonny Zivku
Maplesoft Product Manager, Maple T.A.

Maple T.A. course materials – Statistics (University...

by: jzivku 640

August 20 2014

0 0

Several Maple T.A. users have developed comprehensive sets of question content and assignments to support full courses in Maple T.A. These questions are available through the Maple T.A. Cloud, and we have decided to also post the associated course modules on Maple Primes as an alternative way of accessing this content.

Below you will find a link to the Statistics Maple T.A.. course module developed by the University of Waterloo.

This testing content is freely distributed, and can be used in your own Maple T.A. tests either as-is, or with edits.

The Statistics content is used in introductory statistics courses at the University of Waterloo, and has been used regularly over several years. The over 700 questions are clearly organized by topic, and provide extensive feedback to students.

Topics include:

Basics
Confidence Intervals
Continuous Distribution
Discrete Multivariate
Discrete Probability
Graphical Analysis
Hypothesis Testing
Numerical Analysis for Statistics
Probability
Sampling Distributions

Jonny Zivku
Maplesoft Product Manager, Maple T.A.

Maple T.A. course materials – Calculus 1 and 2

by: jzivku 640

August 15 2014

1 0

Several Maple T.A. users have developed comprehensive sets of question content and assignments to support full courses in Maple T.A. These questions are available through the Maple T.A. Cloud, and we have decided to also post the associated course modules on Maple Primes as an alternative way of accessing this content.

Below you will find a link to the Calculus 1 Maple T.A.. course module developed by the University of Guelph. This course material also forms part of Teaching Calculus with Maple: A Complete Kit, which provides lectures notes, Maple demonstrations, Maple T.A. assignments, and more for teaching both Calculus 1 and Calculus 2.

This testing content is freely distributed, and can be used in your own Maple T.A. tests either as-is, or with edits.

The Calculus 1 course module is designed to accompany the first semester of an introductory honours calculus course. The course is intended primarily for students who need or expect to pursue further studies in mathematics, physics, chemistry, engineering and computer science.

Topics include:

trigonometry including the compound angle formulas
inequalities and absolute values
limits and continuity using rigorous definitions, the derivative and various applications (extreme, related rates, graph sketching)
Rolle's Theorem and the Mean Value Theorem for derivatives
the differential and anti-differentiation
the definite integral with application to area problems
the Fundamental Theorem of Calculus
logarithmic and exponential functions
the Mean Value Theorem for Integrals

The Calculus 2 course module is designed to accompany the second semester of an introductory honours calculus course.

Topics include:

inverse trigonometric functions
hyperbolic functions
L'Hôpital's Rule
techniques of integration
parametric equations
polar coordinates
Taylor and MacLaurin series
functions or two or more variables
partial derivatives
multiple integration

Jonny Zivku
Maplesoft Product Manager, Maple T.A.

Packing of circles: touchstone for global optimize...

by: Markiyan Hirnyk 7759 Products: Maple , Maple Toolboxes

August 13 2014

3 1

I would like to pay attention to a series of applications by Samir Khan
http://www.maplesoft.com/applications/view.aspx?SID=153600
http://www.maplesoft.com/applications/view.aspx?SID=153599
http://www.maplesoft.com/applications/view.aspx?SID=153596
http://www.maplesoft.com/applications/view.aspx?SID=153598
My congratulations to the author on his work well done. New capacities of Global Optimization Toolbox are spectacular. For example, in the first application an optimization
problem in 101 variables under 5050 nonlinear constraints
(other than 202 bounds) is solved.
I think it requires a very powerful comp and much time.
I tried that problem for n=20 with the good old DirectSearch
on my comp (4 GB RAM, Pentium Dual-Core CPU E5700@3GHz) by

soln2 := DirectSearch:-GlobalSearch(rc, {cons1, cons2, rc >= 0,
seq(`and`(vars[i] >= -70, vars[i] <= 70), i = 1 .. 2*n), rc <= 70},
variables = vars, method = quadratic, number = 140, solutions = 1,
evaluationlimit = 20000)

and obtained not so bad rc=69.9609360106765 (whereas www.packomania.com gives rc=58.4005674790451137175957) in about one hour.

Packing_by_DS.mw
For n=50 the memory of my comp cannot allocate calculations or the obtained result by the Search command is far away from the one in packomania.

Pixel Conversion

by: John Dolese 125 Product: Maple 18

August 09 2014

3 1

Here is an example of manipulating an Array of pixels. I chose the x-rite ColorChecker as a model so there would be published results to check my work. A number of details about color spaces have become clear through this exercise. The color adaptation process was modeled by converting betweenXYZ and LMS. Different black points may be selected depending on how close to zero illuminance one would accept as a good model.

I look forward to extending this work to verify and improve the color calibration of my photography. Also some experimentation with demosaicing should be possible.

Initialization

x-rite Colorchecker xyY Matrix

$CCxyY_D50 := Matrix(4, 6, {(1, 1) = Vector(3, {(1) = .4316, (2) = .3777, (3) = .1008}), (1, 2) = Vector(3, {(1) = .4197, (2) = .3744, (3) = .3495}), (1, 3) = Vector(3, {(1) = .2760, (2) = .3016, (3) = .1836}), (1, 4) = Vector(3, {(1) = .3703, (2) = .4499, (3) = .1325}), (1, 5) = Vector(3, {(1) = .2999, (2) = .2856, (3) = .2304}), (1, 6) = Vector(3, {(1) = .2848, (2) = .3911, (3) = .4178}), (2, 1) = Vector(3, {(1) = .5295, (2) = .4055, (3) = .3118}), (2, 2) = Vector(3, {(1) = .2305, (2) = .2106, (3) = .1126}), (2, 3) = Vector(3, {(1) = .5012, (2) = .3273, (3) = .1938}), (2, 4) = Vector(3, {(1) = .3319, (2) = .2482, (3) = 0.637e-1}), (2, 5) = Vector(3, {(1) = .3984, (2) = .5008, (3) = .4446}), (2, 6) = Vector(3, {(1) = .4957, (2) = .4427, (3) = .4357}), (3, 1) = Vector(3, {(1) = .2018, (2) = .1692, (3) = 0.575e-1}), (3, 2) = Vector(3, {(1) = .3253, (2) = .5032, (3) = .2318}), (3, 3) = Vector(3, {(1) = .5686, (2) = .3303, (3) = .1257}), (3, 4) = Vector(3, {(1) = .4697, (2) = .4734, (3) = .5981}), (3, 5) = Vector(3, {(1) = .4159, (2) = .2688, (3) = .2009}), (3, 6) = Vector(3, {(1) = .2131, (2) = .3023, (3) = .1930}), (4, 1) = Vector(3, {(1) = .3469, (2) = .3608, (3) = .9131}), (4, 2) = Vector(3, {(1) = .3440, (2) = .3584, (3) = .5894}), (4, 3) = Vector(3, {(1) = .3432, (2) = .3581, (3) = .3632}), (4, 4) = Vector(3, {(1) = .3446, (2) = .3579, (3) = .1915}), (4, 5) = Vector(3, {(1) = .3401, (2) = .3548, (3) = 0.883e-1}), (4, 6) = Vector(3, {(1) = .3406, (2) = .3537, (3) = 0.311e-1})})$

=

	Convert xyY to XYZ

=

	XYZ D50 to XYZ D65

=

Convert XYZ to Lab (D50 or D65 White Point)

Reference White Point for D50

=

Lab Conversion Constants;

fx_D50 := proc (XYZ) options operator, arrow; piecewise(`ε` < XYZ[1]/X_D50wht, (XYZ[1]/X_D50wht)^(1/3), XYZ[1]/X_D50wht <= `ε`, (1/116)*kappa*XYZ[1]/X_D50wht+4/29) end proc

fy_D50 := proc (XYZ) options operator, arrow; piecewise(`ε` < XYZ[2]/Y_D50wht, (XYZ[2]/Y_D50wht)^(1/3), XYZ[2]/Y_D50wht <= `ε`, (1/116)*kappa*XYZ[2]/Y_D50wht+4/29) end proc

fz_D50 := proc (XYZ) options operator, arrow; piecewise(`ε` < XYZ[3]/Z_D50wht, (XYZ[3]/Z_D50wht)^(1/3), XYZ[3]/Z_D50wht <= `ε`, (1/116)*kappa*XYZ[3]/Z_D50wht+4/29) end proc

XYZ_to_Lab_D50 := proc (XYZ) options operator, arrow; `<,>`(116*fy_D50(XYZ)-16, 500*fx_D50(XYZ)-500*fy_D50(XYZ), 200*fy_D50(XYZ)-200*fz_D50(XYZ)) end proc:

Reference White Point for D65

=

fx_D65 := proc (XYZ) options operator, arrow; piecewise(`ε` < XYZ[1]/X_D65wht, (XYZ[1]/X_D65wht)^(1/3), XYZ[1]/X_D65wht <= `ε`, (1/116)*kappa*XYZ[1]/X_D65wht+4/29) end proc

fy_D65 := proc (XYZ) options operator, arrow; piecewise(`ε` < XYZ[2]/Y_D65wht, (XYZ[2]/Y_D65wht)^(1/3), XYZ[2]/Y_D65wht <= `ε`, (1/116)*kappa*XYZ[2]/Y_D65wht+4/29) end proc

fz_D65 := proc (XYZ) options operator, arrow; piecewise(`ε` < XYZ[3]/Z_D65wht, (XYZ[3]/Z_D65wht)^(1/3), XYZ[3]/Z_D65wht <= `ε`, (1/116)*kappa*XYZ[3]/Z_D65wht+4/29) end proc

XYZ_to_Lab_D65 := proc (XYZ) options operator, arrow; `<,>`(116*fy_D65(XYZ)-16, 500*fx_D65(XYZ)-500*fy_D65(XYZ), 200*fy_D65(XYZ)-200*fz_D65(XYZ)) end proc:

C_XYZ_to_Lab := proc (XYZ, L) options operator, arrow; piecewise(evalb(L = D50), Array([`$`('[`$`('XYZ_to_Lab_D50(XYZ[m, n])', n = 1 .. N)]', m = 1 .. M)]), evalb(L = D65), Array([`$`('[`$`('XYZ_to_Lab_D65(XYZ[m, n])', n = 1 .. N)]', m = 1 .. M)])) end proc

=

Convert XYZ to aRGB (XYZ D50 or D65 to aRGB D65)

XYZ Scaling for aRGB Ymax,Ymin (Ref. Adobe RGB (1998) Color Image Encoding Section 4.3.2.2 and 4.3.8)

White Point (Luminance=160Cd/m^2) D65

Black Point (Luminance=0.5557Cd/m^2) D65

White Point (Luminance=160Cd/m^2) D50

Black Point (Luminance=0.5557Cd/m^2) D50

=

XYZD65_to_aXYZ := proc (XYZ) options operator, arrow; `<,>`((XYZ[1]-XK_D65)*XW_D65/((XW_D65-XK_D65)*YW_D65), (XYZ[2]-YK_D65)/(YW_D65-YK_D65), (XYZ[3]-ZK_D65)*ZW_D65/((ZW_D65-ZK_D65)*YW_D65)) end proc:

XYZD50_to_aXYZ := proc (XYZ) options operator, arrow; `<,>`((XYZ[1]-XK_D50)*XW_D50/((XW_D50-XK_D50)*YW_D50), (XYZ[2]-YK_D50)/(YW_D50-YK_D50), (XYZ[3]-ZK_D50)*ZW_D50/((ZW_D50-ZK_D50)*YW_D50)) end proc:

(ref. Adobe RGB(1998) section 4.3.6.1, Bradford Matrix includes D50 to D65 adaptation)

$M_XYZtoaRGB_D50 := Matrix(3, 3, {(1, 1) = 1.96253, (1, 2) = -.61068, (1, 3) = -.34137, (2, 1) = -.97876, (2, 2) = 1.91615, (2, 3) = 0.3342e-1, (3, 1) = 0.2869e-1, (3, 2) = -.14067, (3, 3) = 1.34926})$

(ref. Adobe RBG(1998) section 4.3.4.1, Bradford Matrix assumes XYZ is D65)

$M_XYZtoaRGB_D65 := Matrix(3, 3, {(1, 1) = 2.04159, (1, 2) = -.56501, (1, 3) = -.34473, (2, 1) = -.96924, (2, 2) = 1.87597, (2, 3) = 0.4156e-1, (3, 1) = 0.1344e-1, (3, 2) = -.11836, (3, 3) = 1.01517})$

aRGB Expansion for 8bits

RGB_to_aRGB := proc (RGB) options operator, arrow; `<,>`(round(255*Norm(RGB[1])^(1/`γa`)), round(255*Norm(RGB[2])^(1/`γa`)), round(255*Norm(RGB[3])^(1/`γa`))) end proc:

Combine Steps

XYZ_to_aRGB := proc (XYZ, L) options operator, arrow; piecewise(evalb(L = D50), Array([`$`('[`$`('RGB_to_aRGB(aXYZ_to_RGB_D50(XYZD50_to_aXYZ(XYZ[m, n])))', n = 1 .. N)]', m = 1 .. M)]), evalb(L = D65), Array([`$`('[`$`('RGB_to_aRGB(aXYZ_to_RGB_D65(XYZD65_to_aXYZ(XYZ[m, n])))', n = 1 .. N)]', m = 1 .. M)])) end proc

Note: The aRGB values published for ColorChecker assume a black point of 0cd/m^2.

=

	Convert XYZ to ProPhoto RGB (D50)

=

	Convert XYZ to sRGB (XYZ D50 or D65 to sRGB D65)

Note: The sRGB values published for ColorChecker assume a black point of 0cd/m^2.

=

Corrections to the original version of theis document;
Make the scaling for a nonzero black point the same for all RGB color spaces.
Clip negative RGB values to zero.
Remove the redundant Array container from matrix multiplications.
Use map in place of the $ to apply a function to each element of an Array.

Pixel_Conversion_B.mw

Interactive Embedded Components in Physics

by: Lenin Araujo Castillo 1790

August 06 2014

2 0

The Interactive Embedded Components in Physics are of great importance today and will be even more in the future. Hereand leave a small tutorial of Embedded Components in Physics applied to physics. I hope that somehow you motives to continue the development of science.

Interactive_Embedded_Components_in_Physics.mw (in spanish)

Ponencia_CRF.pdf

Atte.

Lenin Araujo Castillo

Physics Pure

Computer Science

Stem and Leaf Plot

by: Daniel Skoog 1761 Product: Maple

July 04 2014

2 0

Following @acer 's challenge to create some more examples for the Rosetta Code project, I've put together some code that constructs Stem-And-Leaf plots here.

I've also attached a new mathapp ( StemAndLeafDisplay.mw ) that contains the code as well as an interactive example for Stem-Plots. This MathApp is also viewable online on the MapleCloud here.

This older post may also be of interest for anyone looking to make a stem and leaf plot with decimals.

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