Engineering Mathematics

Mark Willis, Sep 22, 2022

Table of Contents

  1. 1st order differential equations
    1. Solution of a 1st order differential equation with slope field.
    2. Solving a differential equation for an electrical circuit.
    3. The limiting value of a differential equation 02.
    4. The limiting value of a differential equation 01.
    5. Differential equation solution of a mass being projected upwards
    6. Solution of a differential equation by separating the vars
  2. 2nd order differential equations
    1. Second order differential equations damped oscillations
    2. Non-homogeneous 2nd order differential equations
    3. Modelling a bungee jump with a 2nd order differential equation.
    4. The steady state response of a damped second-order system
    5. A system of first-order differential equations
  3. Differential equations using Laplace
    1. Solving a 1st-order differential equation using Laplace transformations
    2. Solving a 2nd-order differential equation using Laplace transformations
    3. Graphical solution to 2nd order differential equation
    4. Graphical solution of a 2nd-order differential equation
    5. Graphs of Laplace transforms
  4. Euler's method
    1. Euler's method for f'(t, x) = x-2t, x(0) = 1
    2. Euler's method for f'(t, x)= xt, x(0) = 2
    3. Euler's method for f'(t, x)=x/(2sqrt(t + x)), x(0.5) = 1
    4. Euler's method for f'(t, x)=(4 - t)/(t + x)
    5. Euler's method for f'(t, x)=1/xt, x(1)=1
    6. Euler's method for f'(t, x)=xt/(x^2+2), x(1)=2
    7. Euler's method for f'(t, x)=1/lnx, x(1)=1.2
    8. Numerical solution of a third-order differential equation 01
    9. Numerical solution of a second-order differential equation
  5. Fourier series
    1. Integrals of periodic functions
    2. Square wave using Fourier terms
    3. Sawtooth wave using Fourier terms
    4. Triangular wave using Fourier terms
    5. Fourier series for a periodic function 01
    6. Fourier series for a periodic function 02
    7. Fourier series for a periodic function 03
    8. Fourier series for a periodic function 04
    9. Fourier series problem
    10. Fourier series for a piecewise periodic function 01
    11. Fourier series for a piecewise periodic function 02
    12. Fourier series for a piecewise periodic function 03
    13. Fourier series for a piecewise periodic function 04
    14. Fourier series for a piecewise periodic function 05
    15. Fourier series for a piecewise function 01
    16. Fourier series for a piecewise function 02
    17. Fourier series for a piecewise function 03
    18. Fourier series for a piecewise function 04
    19. Fourier series for a piecewise function 05
    20. Fourier series for a piecewise function 06
    21. An even function
    22. An odd function
    23. Fourier half-range series
    24. Significance of the constant term in Fourier series
    25. How to change the period of a sine graph
    26. How to change the period of a cosine graph
    27. Fourier series with an arbitrary period of 2
    28. Fourier series with an arbitrary period of 4
    29. Fourier half-range cosine or sine series
  6. Multivariable calculus
    1. The domain of the graph for a surface.
    2. Sketching contour curves of a function of two variables 01.
    3. Sketching contour curves of a function of two variables 02.
    4. The geometrical meaning of partial derivatives.
    5. Finding stationary points of surfaces 01
    6. Finding stationary points of surfaces 02.
    7. Tangent planes and normals to a point introduction.
    8. The tangent plane and normal line to a surface 01.
    9. The tangent plane and normal line to a surface 02.
    10. Cylindrical coordinates.
    11. Spherical coordinates
    12. Triple integral to determine volume
    13. Using a triple integral to determine volume
  7. Linear Algebra
    1. Finding invariant lines under a transformation given by a matrix.
    2. A line of invariant points
    3. Eigenvalues and eigenvectors
    4. Eigenvalues and eigenvectors
    5. Eigenvalues and eigenvectors in 3D

1.

1st order differential equations

  • 1. Solution of a 1st order differential equation with slope field.
  • 2. Solving a differential equation for an electrical circuit.
  • 3. The limiting value of a differential equation 02.
  • 4. The limiting value of a differential equation 01.
  • 5. Differential equation solution of a mass being projected upwards
  • 6. Solution of a differential equation by separating the vars

1.1.

Solution of a 1st order differential equation with slope field.

– Mark Willis

1.2.

–

1.3.

–

1.4.

–

1.5.

–

1.6.

–

2.

2nd order differential equations

  • 1. Second order differential equations damped oscillations
  • 2. Non-homogeneous 2nd order differential equations
  • 3. Modelling a bungee jump with a 2nd order differential equation.
  • 4. The steady state response of a damped second-order system
  • 5. A system of first-order differential equations

2.1.

Second order differential equations damped oscillations

– Mark Willis

2.2.

–

2.3.

–

2.4.

–

2.5.

–

3.

Differential equations using Laplace

  • 1. Solving a 1st-order differential equation using Laplace transformations
  • 2. Solving a 2nd-order differential equation using Laplace transformations
  • 3. Graphical solution to 2nd order differential equation
  • 4. Graphical solution of a 2nd-order differential equation
  • 5. Graphs of Laplace transforms

3.1.

Solving a 1st-order differential equation using Laplace transformations

– Mark Willis

3.2.

–

3.3.

–

3.4.

–

3.5.

–

4.

Euler's method

  • 1. Euler's method for f'(t, x) = x-2t, x(0) = 1
  • 2. Euler's method for f'(t, x)= xt, x(0) = 2
  • 3. Euler's method for f'(t, x)=x/(2sqrt(t + x)), x(0.5) = 1
  • 4. Euler's method for f'(t, x)=(4 - t)/(t + x)
  • 5. Euler's method for f'(t, x)=1/xt, x(1)=1
  • 6. Euler's method for f'(t, x)=xt/(x^2+2), x(1)=2
  • 7. Euler's method for f'(t, x)=1/lnx, x(1)=1.2
  • 8. Numerical solution of a third-order differential equation 01
  • 9. Numerical solution of a second-order differential equation

4.1.

Euler's method for f'(t, x) = x-2t, x(0) = 1

– Mark Willis

4.2.

–

4.3.

–

4.4.

–

4.5.

–

4.6.

–

4.7.

–

4.8.

–

4.9.

–

5.

Fourier series

  • 1. Integrals of periodic functions
  • 2. Square wave using Fourier terms
  • 3. Sawtooth wave using Fourier terms
  • 4. Triangular wave using Fourier terms
  • 5. Fourier series for a periodic function 01
  • 6. Fourier series for a periodic function 02
  • 7. Fourier series for a periodic function 03
  • 8. Fourier series for a periodic function 04
  • 9. Fourier series problem
  • 10. Fourier series for a piecewise periodic function 01
  • 11. Fourier series for a piecewise periodic function 02
  • 12. Fourier series for a piecewise periodic function 03
  • 13. Fourier series for a piecewise periodic function 04
  • 14. Fourier series for a piecewise periodic function 05
  • 15. Fourier series for a piecewise function 01
  • 16. Fourier series for a piecewise function 02
  • 17. Fourier series for a piecewise function 03
  • 18. Fourier series for a piecewise function 04
  • 19. Fourier series for a piecewise function 05
  • 20. Fourier series for a piecewise function 06
  • 21. An even function
  • 22. An odd function
  • 23. Fourier half-range series
  • 24. Significance of the constant term in Fourier series
  • 25. How to change the period of a sine graph
  • 26. How to change the period of a cosine graph
  • 27. Fourier series with an arbitrary period of 2
  • 28. Fourier series with an arbitrary period of 4
  • 29. Fourier half-range cosine or sine series

5.1.

Integrals of periodic functions

– Mark Willis

5.2.

–

5.3.

–

5.4.

–

5.5.

–

5.6.

–

5.7.

–

5.8.

–

5.9.

–

5.10.

–

5.11.

–

5.12.

–

5.13.

–

5.14.

–

5.15.

–

5.16.

–

5.17.

–

5.18.

–

5.19.

–

5.20.

–

5.21.

–

5.22.

–

5.23.

–

5.24.

–

5.25.

–

5.26.

–

5.27.

–

5.28.

–

5.29.

–

6.

Multivariable calculus

  • 1. The domain of the graph for a surface.
  • 2. Sketching contour curves of a function of two variables 01.
  • 3. Sketching contour curves of a function of two variables 02.
  • 4. The geometrical meaning of partial derivatives.
  • 5. Finding stationary points of surfaces 01
  • 6. Finding stationary points of surfaces 02.
  • 7. Tangent planes and normals to a point introduction.
  • 8. The tangent plane and normal line to a surface 01.
  • 9. The tangent plane and normal line to a surface 02.
  • 10. Cylindrical coordinates.
  • 11. Spherical coordinates
  • 12. Triple integral to determine volume
  • 13. Using a triple integral to determine volume

6.1.

The domain of the graph for a surface.

– Mark Willis

6.2.

–

6.3.

–

6.4.

–

6.5.

–

6.6.

–

6.7.

–

6.8.

–

6.9.

–

6.10.

–

6.11.

–

6.12.

–

6.13.

–

7.

Linear Algebra

  • 1. Finding invariant lines under a transformation given by a matrix.
  • 2. A line of invariant points
  • 3. Eigenvalues and eigenvectors
  • 4. Eigenvalues and eigenvectors
  • 5. Eigenvalues and eigenvectors in 3D

7.1.

Finding invariant lines under a transformation given by a matrix.

– Mark Willis

7.2.

–

7.3.

–

7.4.

–

7.5.

–

Information