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Science and Mathematics Simulations UQ

Juan Carlos Ponce Campuzano, Feb 20, 2017

These Science and Mathematics Simulations (SciMS) 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 https://teaching.smp.uq.edu.au/scims/ The worksheets are divided into chapters according to the menu at the site.

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
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
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
Numerical Analysis
1. Riemann sums
2. Legendre polynomials
3. Chebyshev polynomials
4. Polynomial approximation
5. Fourier series
6. Approximation of multivariable functions
Linear Algebra
1. Vector properties and basic operations
2. Systems of linear equations
3. Basic linear transformations
4. Matrix transformations
5. Activity: Quadratic forms
Complex Analysis
1. Basic operations with complex numbers
2. Complex mappings
3. Möbius transformation

Information

1.

Calculus

1.1.

Sequences of real numbers

Consider the sequence

defined by

for

. In the following simulation you can plot a finite number of terms of the sequence. Things to try:

Drag the slider n to explore the values of .
Change the function defining the sequence.
Example: (-1)^n/(sqrt(n+1))
Example: sin(n+1)/(2*n)

1.2.

1.3.

1.4.

1.5.

1.6.

1.7.

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 f(x,y) at point A in

. It also shows the tangent plane at that point. Things to try:

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_x or f_y.
Use right click and drag the mouse to rotate the 3D view or click on View button.

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

Numerical Analysis

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

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

Complex Analysis

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

6.1.

Basic operations with complex numbers

Instructions:

Select an operation (right screen): addition, subtraction, multiplication, division or a linear combination.
Drag the points to change the complex numbers, or use the input boxes (left screen).
Enter your answer in cartesian form to show a geometric representation of the result.

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