Add Math Dynamics to Your WordPress Site

by jaye
Help with the title which describes the Result

Title

The Title field is where you can specify the meaning of the Result field.

The text is informational only and has no hard limit on its length. The text in the Title field is used as the default name for xg2 function definition files that are downloaded.


Help for loading function definitions previoulsy downloaded

Load the Function Definition for an .xg2 File

Click the browse button to select or change the selected Function Definition (.xg2) file to load.


Help for entering the Expression for the Function Definition

Function Definition

The Function Defintition field is where you enter most any mathematical expression

Expressions may represent many functions from a simple constant to polynomials, logrithms, exponents, etc. From a Mathematics standpoint these Expressions, formulas, etc.. are equations where the Result represents the value of the current state of the Function Definition. (Result = Expression) where Expression is a Mathematical Function consisting of constants, variables and intrinsic functions contained as operands within a structure of Math Operators.

TIP: All the spaces in function definition are removed before interpretation by the Math Dynamics MARSHALLING ENGINE.

A number followed by a character is interpreted as follows: (‘1a’ = ‘1 a’ = 1) whereas any character followed by number is interpreted as a variable name, i.e. ‘a12’ or ‘a 12’ is a variable named a12. The Expression field is colloquial in nature and accepts most any combination of letters, numbers and operators. The syntax is uncomplicated and in fact is the same syntax used in computer software. All operations must be explicitly defined with an operator character, +, -, *, /, (, ).

There are some Reserved Words that represent intrinsic functions. These functions include

  1. sin
  2. cos
  3. tan
  4. asin
  5. acos
  6. atan
  7. exp
  8. abs
  9. log
  10. ln

The Math Dynamics COMPUTE ENGINE responds to every change in the Expression field, updating the Result field in real time.


Degrees Radians
Help for trig mode

Trig Mode

Sets a trigonometric function to either calculate the results in degrees or radians.

Help for the Variable Table

Variable List Space

The Variable List Space conains a tabular list of the variables defined in the Expression field.

If there are no variables the Space will appear blank. Variables appear in the order they are encountered in the marshalling process. Each variable is represented as one row in the table. Each row identifies the variable name and its currently assigned value. As you edit the values of the variables in the table the COMPUTE ENGINE updates the Result field in real time.

TIP A Graph of the function will be generated whenever there is an independent variable defined. Independent variables have three values, separated with spaces, in their respective row in the table of variables. For example -3.14 3.14 .01 would generate and plot 628 points in the domain of -pi to +pi. In this case it would be best to set the Trig Mode to Radians.

TIP When the value portion of the variable’s entry is blank, it’s value is zero. Variable names must always begin with a Letter and can be any combination of Letters and Numerals. Variable names are Case Sensitive hence 1/time IS NOT the same as 1/Time.


Help for the result of the evaluation of the function definition

Result

The Result field shows the current value of the state of the Function Definition

A new result is calculated and displayed anytime the value of any of the variables are changed or the trig mode is changed


Help with saving the state of a function definition

Download the Function Definition

Download and save the all of the information about this Function Definition.

This will create and download a .xg2 file named using the Title field. The .xg2 file can then be reloaded using the Choose File and Upload Buttons. The .xg2 file can also be imported into the Math Dynamics Android App.

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Domain
Range

JamworksPro Takes Online Graphing Seriously

by jaye

Jaye at JamworksPro has been busy over the past few months polishing his Flagship Multivariate Algebraic Expression Based Graphing Calculator, Math Dynamics. It’s more than just a graphing calculator. It’s an entire Mathematics Management System.

Warrenton Oregon, Oct 28, 2021— From the JamworksPro R&D Department Comes Math Dynamics, the Online Arithmetic Machine (https://mathdynamics.net). Math Dynamics includes both an Android App as well as it’s Complementary Web App. At the core of Math Dynamics is the ability to write mathematical expressions using most nearly any string of alphanumeric characters as variables.

“Math doesnt have to be cryptic”

a*x^2 + b*x + c
can also be written as
ace*xray^2 + baker*xray + charlie

Math Dynamics keeps it simple with a straight forward approach to variables, constants and the operators used to form mathematical expressions.

If  h is the height measured in feet,  t  is the number of seconds the object has fallen from an initial height h0 with an initial velocity or speed v0 (ft/sec), then the model for height of a falling object is: 

h(t)=−16 * t^2 + v0 * t + h0

“Free Form Variable Names make more sense in the physical world”

height(elapsedTime) = -16 * elapsedTime^2 + initialVelocity * elapsedTime + initialHeight

Mathematical Expressions are packaged in Math Dynamics as Function Definition Cards. The Variables are listed in a table that allows their values to be edited. Any one of the variables can be used as an independent variable to plot the graph of the function. By entering start, end and increment values in that order will define that variable as independent and will plot a graph accordingly. Function Definition Cards can be named and saved as .xg2 files for portablility. These .xg2 files can be opened by both the Web App as well as the Android App.

This Function Definition Card demonstrates the straight forward approach to assigning meaningful names to the variables.
This Function Definition Card defines the Quadratic Function in terms that have coefficients with free form names. Note that the value of the ‘xray’ term defines the x axis limits as -3 to 7, an increment of .001 and y axis limits as -10 to 10.

Collections of Function Cards are packaged in the Math Dynamics Pallet (https://mathdynamics.net/pallet.html). Both the Android App and the Web App have a Pallet. Function Definition Cards are easily added to or removed from the Pallet. The Pallet in the Android App supports Function Definition Cards that each have a self contained graph while the Pallet in the Web App supports a single graph on which any of the Function Definition Cards may use to plot its graph.

The Pallet in the Web App is very powerful. It supports plotting multiple functions on the same graph. Each plot can have a different color and each plot can also be hidden. Like Function Definition Cards, Pallets can be named and can be downloaded and saved for portability.

The Web App Pallet supports up to 26 custom functions. Each function can be plotted on the common graph. The visibility of the graph can be turned on or off. Each graph can have a custom color assigned to it.
The Math Dynamics Function Pallet provides a way to graph multiple functions on the same graph
This Graph of the Maclaurin Series to calculate the sin of an angle demonstrates the ability to plot multiple functions on the same graph.

Each Function Definition Card in the Math Dynamics Web App Function Pallet is assigned its own three letter moniker. A Function moniker always begins with FN and the third letter is determined by the position of Function Definition Card in the Pallet. For example the first Function Definition Card will have the moniker FNA and the second, FNB. The Math Dynamics Web App Pallet supports 26 of these function monikers, FNA thru FNZ.

These monikers allow each Function Definition in the Pallet to used as an intrinsic function such as sin(), cos(), sqrt(), etc.. in any other Function Definition Card. The argument of the calling Function will be passed to the called Functions independent variable.

The Math Dynamics Web App Function Pallet is designed for training and research. The Web App contains an editor in the left panel that is used to write details and instructions about the Pallet, allowing entire tutorials to be created. Using the Pallet Download and Load features provides for great portability.

This Math Dynamics Function Pallet provides a tutorial on the fundamental concepts of Differential Calculus.

Jaye Mosier, owner of JamworksPro is an Independent Software Developer specializing in small business web development as well as android mobile app development. Complex back end data systems interfaced with simple and elegant GUI design is one of Jaye’s many specialties.

Jaye (https://jaye.jamworkspro.com) has published many software title over the years including FileNotes, TRAX, Coupon Cache and others as well as the most recent projects, Math Dynamics, Vibrology App and The Pothole Project.

Exploring the Maclaurin Series Expansion sin(x)

by jaye

sin(x) = \(x – x^3/3! + x^5/5! – x^7/7! + x^9/9! – \)…

The sin(x) is plotted in black.

Lets break down the amount of work we need to do depending on size of the angle.

2 Terms are needed for angles 0 through .9 radians

3 Terms are needed for angles >.9 and <1.57

4 Terms are needed for angles >1.57 and <2.5 radians

5 Terms are needed for angles >2.5 and <3.3 radians

6 Terms are needed for angles >3.3 and <4 radians

7 Terms are needed for angles >4 and <4.5 radians

8 Terms are needed for angles >4.5 and <5.5 radians

9 Terms are needed for angles >5.5 and <=2*pi radians

This can be A LOT of WORK with pencil and paper.

Fortunately, the sin of any (A)NGLE > 2*pi will be identical to the sin of an (a)ngle less than or equal to 2*pi.

So for example, given an (A)ngle of 31.72 radians its corresponding (a)ngle will be .304

n = int(A/(2*pi))
a = A-2*n*pi

Given A = 31.72 then n = 5
hence a = 31.72-(2*5*pi)= .304

GREAT!! we only need 2 Terms x – x^3/fact(3)

Click to explore this Interactive Pallet
Math Dynamics Interactive Function Pallet demonstrating the Maclaurin Expansion Series for the Sin(x)

This interactive Math Dynamics Pallet demonstrates how the Maclaurin Expansion Series can be used to calculate the sin of any angle

A Quadratic Deep Dive into Real Roots

by jaye

The Math Dynamics Multivariable Algebra Expression Based Graphing Calculator

Graph of the Quadratic Equation
This Math Dynamics Function Card will generate a plot from -10 to 10 in increments of .001 and limit the Y axis from 10 to -10

Complete the Square method is an algorithm to find the Roots of a Quadratic Equation both Real and Complex.  This exercise focuses on using the “Complete the Square” method to calculate the Real Roots so it only makes sense to use a Quadratic Function that has two real roots.

We start with a Quadratic Function defined as:

\( -3*x^2 + 2*x + 4 \)

The plot of this function visually demonstrates that this Quadratic Function has 2 real roots. 

The exact values of these two real roots can be calculated using the “Complete the Square” method.  This requires two things:

  1. \( -3*x^2 + 2*x + 4 = 0\)
  2. The \(a*x^2\) term requires that a = 1, not -3
Quadratic where the coefficient of the x^2 term is always 1
This Math Dynamics Function Card produces a Quadratic Equation that factors out the coefficient a ensuring it is always 1

To resolve the second requirement requires that both sides of the quadratic equation be multiplied by -(1/3) such that the new Quadratic Function is:

\( x^2 -(2/3)*x -(4/3) = 0\)

Which produces a different graph.  This new graph shares the same roots as the original graph.

I found this to be a quite interesting property.  It appears though that all multiples of a Quadratic Function have equal roots

Visual evidence that all multiples of any quadratic function have the same roots
This Math Dynamics Card defines a function that can generate any multiple of any Quadratic Function
Math Dynamics Function Pallet exploring the Quadratic Equation
Explore the Quadratic Equation and Using the Complete the Square Method to find the roots.

JamworksPro Introduces Math Dynamics™ for Android

by jaye

Math Dynamics, the Multivariate Algebraic Expression Based Graphing Calculator

Math Dynamics is more than a calculator. It’s a platform for managing math intensive information. From simple unit conversions to advanced algebraic and trigonometric analysis. Math Dynamics allows the User to craft custom Function Definitions from many discipline’s such as Statistics, Marketing, Engineering, etc.

Its a tool for Math. Another instrument that you can add to your tool chest that gives you an edge when it comes to everyday activities that demand a quick calculation.

Use Calculator mode for a quick calculation. Calculations can range from simple arithmetic to multivariate formulas like Cost per Unit, Miles per Gallon, etc..

 

Any formula created in the Calculator can be saved to the Pallet.

The Pallet is a collection of Function Definition Cards. Each card contains the function definition and a list of all of the variables defined and it’s evaluated result. Users can quickly evaluate any Function Definition simply by editing the values of the variables in the table. The result is evaluated in real time as you change the values of the variables.

With Math Dynamics you can share your function definitions with other users. Export and Import functions allow the Users to save and share any of the Function Definition’s in the Pallet as a Math Dynamics XML file. These files also include the values of the variables, the result and the Trig Mode at the time of Export.

A Treatise on Trigonometry

by jaye

Trigonometry is based on the fundamental principle of ONE length.  The length in this context is immaterial because with Trigonometry we only need be concerned about ONE.  For example if you draw a circle it is ONE circle no matter how big or small.  Because it is a circle it has only ONE center with a constant distance to the edge of the circle.  This constant distance is ONE and is called the Radius.

The Unit Circle has a radius of ONE and sweeps out an Angle of ONE Radian
The Unit Circle is fundamental to Trigonometry. Any circle has a constant Circumference that is exactly 3.14159… times its Diameter. The “units”, i.e. inches, centimeters, … are immaterial. Its radius is simply ONE unit.

Simple so far, just a circle with a center and a radius of ONE.  Now consider the radius as it sweeps through a small region of the circle forming a triangle with two sides of an equal length ONE  and forming an angle at the apex of the triangle.  An Angle is a unit that specifies the amount of the sweep of the radius that formed the triangle.

At this point we are finished with the circle.  It is no longer needed.  Also, our Triangle can now be moved and reoriented any way we wish.  It is imperative though that the lengths of the two radii forming the triangle remain equal and ONE.  The angle is most visually apparent when one of the sides of the Triangle lies along a horizontal axis.  At this point a perpendicular line can be drawn between the tip of one of radii and intersecting the other radii.  The perpendicular line intersects the radii  such that the length between the apex (angle)  and the intersection is a ratio of ONE.

Now, this is where it gets complicated.  Since the triangle has been conveniently aligned with one of the radii along the horizontal axis it can be said that the SIN is the distance (as a percentage of ONE) along a corresponding VERTICAL AXIS between the TIP of one radii and a HORIZONTAL AXIS.  Likewise the COS is the distance along a HORIZONTAL AXIS  between the TIP of one radii and a VERTICAL AXIS.

Explore things like; How many terms do I need to get an accurate calculation for any angle.   Find out with the  Math Dynamics Multivariate Algebraic Expression Based Graphing Calculator.

Click here for the live interactive Pallet

Math Dynamics Graping Calculator Pallet graphs sin overlayed with series expansions
Math Dynamics Pallet explores calculating sins and cosines from series expansions