Science and Mathematics Simulations UQ juan

Juan Carlos Ponce Campuzano, Ronan Downes, Mar 16, 2020

These [b]Science and Mathematics Simulations (SciMS)[/b] and associated teaching packages were developed to assist student learning at undergraduate level at the University of Queensland. This book contains all the simulations designed with GeoGebra from [url]https://teaching.smp.uq.edu.au/scims/[/url] The worksheets are divided into chapters according to the menu at the site.

Table of Contents

  1. Calculus
    1. Sequences of real numbers
    2. Representation of series
    3. Derivative of a function
    4. Riemann sums
    5. Fundamental Theorem of Calculus
    6. Relative velocity: Boat problem
    7. Elastic collisions
  2. Advanced Calculus
    1. Partial derivatives and Tangent plane
    2. Directional derivatives and Gradient
    3. Line integral for planar curves
    4. Vector Fields in 2D
    5. Vector fields 3D
    6. Line integrals of vector fields: Work & Circulation
    7. Line integrals of vector fields: Flux
    8. Basic examples of velocity fields
    9. Parabolic transformation
    10. Polar transformation
  3. Applied Mathematical Analysis
    1. Slope fields of ordinary differential equations
    2. Euler's Method
    3. Lotka-Volterra model
    4. Undamped Forced Oscillations
    5. Forced Vibrations With Damping
    6. Fourier series
    7. Heat equation
    8. Wave equation
    9. Wave Equation on a Half Line
    10. Wave equation: d'Alembert's formula
  4. Numerical Analysis
    1. Riemann sums
    2. Legendre polynomials
    3. Chebyshev polynomials
    4. Polynomial approximation
    5. Fourier series
    6. Approximation of multivariable functions
  5. Linear Algebra
    1. Vector properties and basic operations
    2. Systems of linear equations
    3. Basic linear transformations
    4. Matrix transformations
    5. Activity: Quadratic forms
  6. Complex Analysis
    1. Basic operations with complex numbers
    2. Complex mappings
    3. Möbius transformation

1.

Calculus

  • 1. Sequences of real numbers
  • 2. Representation of series
  • 3. Derivative of a function
  • 4. Riemann sums
  • 5. Fundamental Theorem of Calculus
  • 6. Relative velocity: Boat problem
  • 7. Elastic collisions

1.1.

Sequences of real numbers

Consider the sequence [math]f:\mathbb{N}\rightarrow\mathbb{R}[/math] defined by [math]f\left(n\right)=a_n[/math] for [math]n\in\mathbb{N}[/math]. In the following simulation you can plot a finite number of terms of the sequence.[br][br][b]Things to try:[/b][br][list][*]Drag the slider [b]n[/b] to explore the values of [math]a_n[/math].[/*][*]Change the function defining the sequence.[b] [/b][/*][*][b]Example: [/b](-1)^n/(sqrt(n+1))[/*][*][b]Example: [/b]sin(n+1)/(2*n)[/*][/list]
– Juan Carlos Ponce Campuzano

1.2.

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1.3.

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1.4.

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1.5.

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1.6.

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1.7.

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2.

Advanced Calculus

  • 1. Partial derivatives and Tangent plane
  • 2. Directional derivatives and Gradient
  • 3. Line integral for planar curves
  • 4. Vector Fields in 2D
  • 5. Vector fields 3D
  • 6. Line integrals of vector fields: Work & Circulation
  • 7. Line integrals of vector fields: Flux
  • 8. Basic examples of velocity fields
  • 9. Parabolic transformation
  • 10. Polar transformation

2.1.

Partial derivatives and Tangent plane

This simulation shows the geometric interpretation of the partial derivatives of [i]f[/i]([i]x,y[/i]) at point A in [math]\mathbb{R}^2[/math]. It also shows the tangent plane at that point.[br][br]Things to try:                [list][*]Drag the point A in the xy-plane or type specific values on the boxes.              [/*][*]Select the object you want to show: Tangent plane, f[sub]x[/sub] or f[sub]y[/sub].           [/*][*]Use right click and drag the mouse to rotate the 3D view or click on View button.[/*][/list]
– Juan Carlos Ponce Campuzano

2.2.

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2.3.

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2.4.

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2.5.

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2.6.

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2.7.

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2.8.

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2.9.

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2.10.

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3.

Applied Mathematical Analysis

  • 1. Slope fields of ordinary differential equations
  • 2. Euler's Method
  • 3. Lotka-Volterra model
  • 4. Undamped Forced Oscillations
  • 5. Forced Vibrations With Damping
  • 6. Fourier series
  • 7. Heat equation
  • 8. Wave equation
  • 9. Wave Equation on a Half Line
  • 10. Wave equation: d'Alembert's formula

3.1.

Slope fields of ordinary differential equations

– Juan Carlos Ponce Campuzano

3.2.

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3.3.

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3.4.

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3.5.

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3.6.

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3.7.

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3.8.

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3.9.

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3.10.

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4.

Numerical Analysis

  • 1. Riemann sums
  • 2. Legendre polynomials
  • 3. Chebyshev polynomials
  • 4. Polynomial approximation
  • 5. Fourier series
  • 6. Approximation of multivariable functions

4.1.

– Juan Carlos Ponce Campuzano

4.2.

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4.3.

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4.4.

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4.5.

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4.6.

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5.

Linear Algebra

  • 1. Vector properties and basic operations
  • 2. Systems of linear equations
  • 3. Basic linear transformations
  • 4. Matrix transformations
  • 5. Activity: Quadratic forms

5.1.

Vector properties and basic operations

– Juan Carlos Ponce Campuzano

5.2.

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5.3.

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5.4.

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5.5.

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6.

Complex Analysis

  • 1. Basic operations with complex numbers
  • 2. Complex mappings
  • 3. Möbius transformation

6.1.

Basic operations with complex numbers

Instructions:
[list][*]Select an operation (right screen): addition, subtraction, multiplication, division or a linear combination.[br][/*][*]Drag the points to change the complex numbers, or use the input boxes (left screen).[br][/*][*]Enter your answer in cartesian form to show a geometric representation of the result.[/*][/list][br][br]
– Juan Carlos Ponce Campuzano

6.2.

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6.3.

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