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sowiso logo Advanced Calculus

Open course material Advanced Calculus offered by KdVI, SMASH and TLC-FNWI

Authors: André Heck, Marthe Schut

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Course content
Complex numbers
Known numbers
THEORY
T
1.
Known numbers
THEORY
T
2.
Adjunction of a square root
PRACTICE
P
3.
Computing in ℚ(√2) and ℚ(√3)
3
Construction of complex numbers and complex arithmetic
THEORY
T
1.
imaginary numbers
PRACTICE
P
2.
Calculating with imaginary numbers
6
THEORY
T
3.
Complex numbers: addition and multiplication
PRACTICE
P
4.
Adding and multiplying complex numbers
7
THEORY
T
5.
Conjugate, modulus, and division of complex numbers
PRACTICE
P
6.
Working with conjugate and norm, plus division
6
THEORY
T
7.
Properties of conjugate and norm
PRACTICE
P
8.
Deriving properties of complex numbers
3
The complex plane
THEORY
T
1.
Complex numbers as points in a plane
PRACTICE
P
2.
Drawing point in the complex plane
3
THEORY
T
3.
Geometric objects via complex numbers
PRACTICE
P
4.
Working with circles in the complex plane
5
THEORY
T
5.
Complex numbers as vectors
PRACTICE
P
6.
Computing with vectors in the complex plane
6
THEORY
T
7.
Complex numbers on the unit circle
PRACTICE
P
8.
Computing an argument
2
THEORY
T
9.
Euler's formula
PRACTICE
P
10.
Computing with imaginary e-powers
4
THEORY
T
11.
de Moivre's formula
PRACTICE
P
12.
Deriving a trigonometric formula
1
THEORY
T
13.
Polar coordinates
PRACTICE
P
14.
Computing with polar coordinates
11
THEORY
T
15.
Multiplication in polar format
PRACTICE
P
16.
Computing in polar form
4
Complex functions
THEORY
T
1.
Complex linear functions
PRACTICE
P
2.
Computing with complex linear functions
4
THEORY
T
3.
Complex polynomial functions
PRACTICE
P
4.
Computing with complex polynomial functions
2
THEORY
T
5.
Complex exponential functions
PRACTICE
P
6.
Computing with complex exponential functions
5
THEORY
T
7.
Complex trigonometric functions
PRACTICE
P
8.
Computing with complex trigonometric functions
2
THEORY
T
9.
The complex logarithm
PRACTICE
P
10.
Computing with the complex logarithm
4
Roots and polynomials
THEORY
T
1.
Complex square roots
PRACTICE
P
2.
Extracting square roots
3
THEORY
T
3.
Complex cube roots
PRACTICE
P
4.
Extracting cube roots
2
THEORY
T
5.
n-th roots of complex numbers
PRACTICE
P
6.
Computing with n-th roots
1
THEORY
T
7.
Solving quadratic equations in ℂ
THEORY
T
8.
The quadratic-formula
PRACTICE
P
9.
Solving quadratic equations
3
Fourier series
Introduction
THEORY
T
1.
Fourier's law of heat conduction
THEORY
T
2.
What is a Fourier series?
THEORY
T
3.
Calculating a Fourier sine series
THEORY
T
4.
Example of a square wave
THEORY
T
5.
Calculating a Fourier cosine series
PRACTICE
P
6.
Practice with determining a Fourier cosine series
2
The Fourier series of an arbitrary function
THEORY
T
1.
The overall concept
THEORY
T
2.
The frequency-amplitude spectrum
THEORY
T
3.
Periodic functions with an arbitrary period
PRACTICE
P
4.
Practice with determining a Fourier cosine series
2
The complex Fourier series
THEORY
T
1.
The complex Fourier series
THEORY
T
2.
Fourier integrals
PRACTICE
P
3.
Computing a Fourier series
1
Functions of several variables
Basic concepts and visualization
THEORY
T
1.
Basic concepts
PRACTICE
P
2.
Computing function values
2
PRACTICE
P
3.
Computing the domain and range of a function
3
THEORY
T
4.
Isolation of a variable
PRACTICE
P
5.
Converting a relation into a functional relationship
1
Visualisation of functions of two variables
THEORY
T
1.
Graphs and coordinate curves
THEORY
T
2.
3D Graphics in Matlab
THEORY
T
3.
3D Graphics in Python
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THEORY
T
4.
Level curves and contour plots
PRACTICE
P
5.
Level curves and contour plots
3
THEORY
T
6.
Contour plots in Matlab
THEORY
T
7.
Contour plots in Python
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THEORY
T
8.
Matlab exercises: 3D graphics
THEORY
T
9.
Worked-out solutions of the Matlab exercises: 3D graphics
THEORY
T
10.
Python exercises: 3D graphics
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THEORY
T
11.
Worked-out solutions of the Python exercises: 3D graphics
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Partial derivatives
THEORY
T
1.
First partial derivatives
PRACTICE
P
2.
Computing first partial derivatives
6
THEORY
T
3.
Higher partial derivatives
PRACTICE
P
4.
Computing higher partial derivatives
10
THEORY
T
5.
Chain rules
PRACTICE
P
6.
Chain rules
5
Tangent vector and tangent plane
THEORY
T
1.
Tangent vectors
THEORY
T
2.
An equation of a tangent plane
PRACTICE
P
3.
Finding tangent planes
4
Total differential and Taylor approximation
THEORY
T
1.
The total differential
THEORY
T
2.
Taylor approximations
PRACTICE
P
3.
Finding a quadratic approximation
1
THEORY
T
4.
Propagation of error
PRACTICE
P
5.
Estimating an error
1
Gradient
THEORY
T
1.
Directional derivative
PRACTICE
P
2.
Directional derivative
2
THEORY
T
3.
The gradient
PRACTICE
P
4.
Computing the gradient of a function
3
Critical points
THEORY
T
1.
Introduction
THEORY
T
2.
Critical points
PRACTICE
P
3.
Finding critical points
2
THEORY
T
4.
Maximum, minimum, and saddle point
THEORY
T
5.
Criteria for maximum, minimum, and saddle point
PRACTICE
P
6.
Classifying a critical point
8
Lagrange multipliers
THEORY
T
1.
The method of Lagrange multipliers (examples)
THEORY
T
2.
The method of Lagrange multipliers (general)
PRACTICE
P
3.
The method of Lagrange multipliers
2
Multiple integrals
Double integrals
THEORY
T
1.
Double integrals as volume
THEORY
T
2.
Double integrals approximated with a Riemann sum
PRACTICE
P
3.
Double integrals approximated with a Riemann sum
1
THEORY
T
4.
Double integrals as iterated integrals
PRACTICE
P
5.
Double integrals as iterated integrals
9
THEORY
T
6.
Double integrals over a bounded, but non-rectangular region
PRACTICE
P
7.
Double integrals over a bounded, but non-rectangular region
7
THEORY
T
8.
Properties of double integrals
PRACTICE
P
9.
Properties of double integrals
2
THEORY
T
10.
Improper double integrals
PRACTICE
P
11.
Improper double integrals
4
THEORY
T
12.
Double integrals in polar coordinates
PRACTICE
P
13.
Double integrals in polar coordinates
4
THEORY
T
14.
Change of variables in double integrals
PRACTICE
P
15.
Change of variables in double integrals
4
Triple integrals
THEORY
T
1.
Triple integrals in Cartesian coordinates
PRACTICE
P
2.
Triple integrals in Cartesian coordinates
5
THEORY
T
3.
Change of variables in triple integrals: spherical and cylindrical coordinates
PRACTICE
P
4.
Change of variables in triple integrals: spherical and cylindrical coordinates
4
Applications of multiple integrals
THEORY
T
1.
Calculation of the volume of a 3-D sub-region
PRACTICE
P
2.
Calculation of the volume of a 3-D sub-region
2
THEORY
T
3.
Calculation of the area of a 2D-region
PRACTICE
P
4.
Calculation of the area of a 2D-region
5
THEORY
T
5.
Average value of a function of 2 or 3 variables
PRACTICE
P
6.
Average value of a function of 2 or 3 variables
2
THEORY
T
7.
Centre of gravity, static moments and moments of inertia
PRACTICE
P
8.
Centre of gravity, static moments and moments of inertia
8
THEORY
T
9.
Application of triple integrals in quantum chemistry
PRACTICE
P
10.
Application of triple integrals in quantum chemistry
1
Ordinary differential equations
Introduction
THEORY
T
1.
What is a differential equation?
THEORY
T
2.
Notation for ODEs
THEORY
T
3.
Terminology
THEORY
T
4.
Types of first-order dynamical systems
PRACTICE
P
5.
Determining the type and order of an ODE
16
THEORY
T
6.
From a differential equation to a function
THEORY
T
7.
From a function to a differential equation
THEORY
T
8.
Additional conditions for a differential equation
PRACTICE
P
9.
Solving an initial value problem or a boundary value problem
3
Separable differential equations
THEORY
T
1.
Integrating a function
THEORY
T
2.
Solving by separation of variables
THEORY
T
3.
Example 1
THEORY
T
4.
Example 2: Exponential growth
THEORY
T
5.
Example 3: Limited exponential growth
THEORY
T
6.
Example 4: Logistic growth
THEORY
T
7.
Overview of growth models
PRACTICE
P
8.
Solving separable ODEs in steps
15
PRACTICE
P
9.
Practising more with solving separable ODEs
2
THEORY
T
10.
Application: Cooling and heating
THEORY
T
11.
Separable ODEs in chemical reaction kinetics
THEORY
T
12.
Exact solutions of an ODE with MATLAB
THEORY
T
13.
Exact solutions of an ODE with Python
THEORY
T
14.
Exact solutions of an ODE with R
Solving ODEs by an integrating factor
THEORY
T
1.
Known examples of ODEs solved by an integrating factor
THEORY
T
2.
An integrating factor for y'=p·y+q·t and y'=p(t)·y
THEORY
T
3.
A non-homogeneous first-order linear differential equation
THEORY
T
4.
An example with 1/t as integrating factor
PRACTICE
P
5.
Practising with an integrating factor
2
THEORY
T
6.
Application: pharmacokinetics of an orally administered drug
THEORY
T
7.
Application: a cascade of first-order chemical reactions
THEORY
T
8.
Linear ODEs of high order
Slope field and solution curves
THEORY
T
1.
Slope field
THEORY
T
2.
'Go with the flow'
THEORY
T
3.
Interactive computer version of a slope field
THEORY
T
4.
Slope field vs direction field
PRACTICE
P
5.
Sketching a slope field and solution curves
7
PRACTICE
P
6.
Working with slope fields
6
THEORY
T
7.
The forward Euler method
THEORY
T
8.
More about the Euler method
THEORY
T
9.
Behaviour of solutions
THEORY
T
10.
Existence and uniqueness of solutions
Slope field and solution curves with Matlab
THEORY
T
1.
Drawing a slope field
[MATLAB task]
THEORY
T
2.
Drawing a slope field
[MATLAB worked-out solution]
THEORY
T
3.
Drawing a slope field with the quiver function in MATLAB
THEORY
T
4.
Drawing a slope field with integral curves [MATLAB task]
THEORY
T
5.
Drawing a slope field with integral curves [MATLAB worked-out solution]
THEORY
T
6.
Forward Euler method
[MATLAB task]
THEORY
T
7.
Forward Euler method
[MATLAB worked-out solution]
THEORY
T
8.
Numerically solving a differential equation in MATLAB
Slope field and solution curves with Python
THEORY
T
1.
Drawing a slope field
[Python task]
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THEORY
T
2.
Drawing a slope field
[Python worked-out solution]
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THEORY
T
3.
Drawing a slope field with the quiver function in Python
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THEORY
T
4.
Drawing a slope field with integral curves [Python task]
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THEORY
T
5.
Drawing a slope field with integral curves [Python worked-out solution]
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THEORY
T
6.
Forward Euler method
[Python task]
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THEORY
T
7.
Forward Euler method
[Python worked-out solution]
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THEORY
T
8.
Numerically solving a differential equation in Python
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Slope field and solution curves with R
THEORY
T
1.
Drawing a slope field [R task]
THEORY
T
2.
Drawing a slope field [R worked-out solution]
THEORY
T
3.
Drawing a slope field with some integral curves [R task]
THEORY
T
4.
Drawing a slope field with some integral curves
[R worked-out solution]
THEORY
T
5.
Drawing a slope field of an autonomous ODE with the flowField function
THEORY
T
6.
Drawing a slope field of an autonomous ODE with integral curves in R
THEORY
T
7.
Forward Euler method [R task]
THEORY
T
8.
Forward Euler method [R worked-out solution]
Asymptotics and stability
THEORY
T
1.
Asymptotics
PRACTICE
P
2.
Finding an asymptotic solution
3
THEORY
T
3.
Stability
THEORY
T
4.
Stability analysis based on sign patterns
THEORY
T
5.
Stability analysis by local linearisation
PRACTICE
P
6.
Stability analysis
9
THEORY
T
7.
Pencil-and-paper exercises on stability analysis
Bifurcations
THEORY
T
1.
Bifurcations and bifurcation diagram
THEORY
T
2.
Saddle-node bifurcation
THEORY
T
3.
Transcritical bifurcation
THEORY
T
4.
Pitchfork bifurcation
PRACTICE
P
5.
Bifurcations and bifurcation diagrams
5
THEORY
T
6.
Additional exercises about bifurcations and bifurcation diagrams
THEORY
T
7.
Bistability and hysteresis
THEORY
T
8.
Application: Spruce Budworm outbreaks
THEORY
T
9.
Application: stability analysis of a 1-dimensional neuron model
Second-order linear ODEs with constant coefficients
THEORY
T
1.
Introduction
THEORY
T
2.
Homogeneous linear ODEs of order 2 with constant coefficients
THEORY
T
3.
Positive discriminant
THEORY
T
4.
Discriminant equal to zero
THEORY
T
5.
Negative discriminant
PRACTICE
P
6.
Solving homogeneous 2nd order linear ODEs with constant coefficients
3
THEORY
T
7.
Application: Vibrations
THEORY
T
8.
Systems of coupled first-order linear ODEs
Systems of differential equations
Linear systems of differential equations
THEORY
T
1.
Uncoupled autonomous differential equations
PRACTICE
P
2.
Uncoupled autonomous differential equations
9
THEORY
T
3.
Coupled autonomous differential equations
THEORY
T
4.
A qualitative phase portrait
THEORY
T
5.
Qualitative exploration of stability
THEORY
T
6.
Pencil-and-paper assignments: a qualitative study of stability
THEORY
T
7.
From a second-order ODE to a linear system of first-order ODEs
THEORY
T
8.
From a linear system of ODEs to a homogeneous 2nd-order ODE
THEORY
T
9.
Linear algebra approach to solving linear systems of differential equations
THEORY
T
10.
Reflection on the solution method
THEORY
T
11.
Two different real eigenvalues
THEORY
T
12.
Drawing a phase portrait in MATLAB
THEORY
T
13.
Drawing a phase portrait in R
THEORY
T
14.
Pencil-and-paper exercise set 1
THEORY
T
15.
One real eigenvalue
THEORY
T
16.
Pencil-and-paper exercise set 2
THEORY
T
17.
Complex eigenvalues
THEORY
T
18.
Pencil-and-paper exercise set 3
THEORY
T
19.
Classification of stability
THEORY
T
20.
Pencil-and-paper exercise set 4
THEORY
T
21.
Pencil-and-paper exercise set 5
Non-linear differential equations
THEORY
T
1.
The phase plane
THEORY
T
2.
Jacobian matrix
THEORY
T
3.
The chain rule
THEORY
T
4.
Analysis near singularities
THEORY
T
5.
Worked-out example: an uncoupled system
THEORY
T
6.
Second worked-out example
THEORY
T
7.
Action potentials in nerve cells: the Fitzhugh-Nagumo model
THEORY
T
8.
Pencil-and-paper exercise set
THEORY
T
9.
A higher-dimensional example: the SIR model
Simulations of single neuron models (implemented in EjsS)
THEORY
T
1.
Hodgkin-Huxley model
THEORY
T
2.
Krinky-Kokoz-Rinzel model
THEORY
T
3.
Basic Wilson model
THEORY
T
4.
Full Wilson model
THEORY
T
5.
FitzHugh-Nagumo model
THEORY
T
6.
The Izhikevich model
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Open course material Advanced Calculus offered by KdVI, SMASH and TLC-FNWI

Authors: André Heck, Marthe Schut

Full access via UvAnetID

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