Lecture 1: part
1 part
2
(In Part 1, the mathematics starts at 12:55. The organizational
matters are somewhat different this year. At 10:10, there is an error:
The homework groups should consist of 4 students.)
Content: linear second order ordinary
differential equations (ODE) with constant coefficients I
Material: [A] 3.7
Concepts: damped / non-damped oscillations, linear ODEs of second
order with constant coefficients, general solution
Abilities: solving linear second order ODEs with constant
coefficients
Exercises: [A] 3.7: 3, 5, 7, 9, 11, 13, 15, 17, 18, 31, 32,
33, 34
Lecture 2: part
1 part
2
(Sorry for the error on the sheet on the ansatz methods. The corrected
version is here. In Part 1 at 12:30,
a factor i is missing in the numbers e^(+-pi/3 * i).
)
Content: linear second order
ordinary differential equations (ODE) with constant coefficients II.
Systems of linear ODEs with constant coefficients I
Material: [A] 19.6, [L] 6.2,
Sheets,
extra material on the general case
Concepts: free / forced oscillation, resonance, homogeneous system
of linear ODEs with constant coefficients, its general solution,
initial value problem
Abilities: solving (initial value problems for) inhomogeneous linear
second order ODEs with constant coefficients by variation of
parameters or
ansatz methods,
solving homogeneous systems of linear ODEs with constant
coefficients in the diagonalizable case
Problems:
[A] 18.6. 1, 3, 7, 11, 20
[L] 6.2 1d,e, 2b, 3
Lecture 3: part
1 part
2
Content: Systems of linear ODEs with constant coefficients
II, Laplace transform I
Material: [A] 20.1, (9th edition 18.7) (Mind the wrong sign in the
definition of "exponential order".)
Sheets, table,
Video
(Khan Academy, first in a series), survey
Concepts: homogeneous / inhomogeneous system of linear ODEs with
constant coefficients, particular solution, Laplace transform
Abilities: solving (initial value problems for) systems of linear
ODEs with constant coefficients in the diagonalizable case,
calculation of Laplace transforms (directly and via general rules)
Exercises Solutions
Lecture 4: part
1 part
2
(Sorry for the confusion around the Laplace transform of a
time-shifted function. Read here what
should have been on the board.)
Content: Laplace transform II
Material: [A] 20.1, (9th edition 18.7),
survey,
table
Concepts: (inverse) Laplace transform
Abilities: Applying Laplace transform when solving initial
value problems for (systems of) linear ODEs with constant
coefficients
Exercises
Solutions
Some internet material on the
Laplace transform:
Homework Set 2
Lecture
5:
(The recording of the lecture starts at 36:20. Unfortunately, the
first part of the lecture is missing.
Here
is the first part of the corresponding lecture from 2018 )
Content: Functions of several variables
Material: [A] 13.1, 2, 10.5 (in part) 10.1-4 (recap of 2WCB0)
Concepts: function of several variables, domain, image, graph,
level line / surface, limit, continuity,
Abilities: for functions of 2 or 3 variables: in simple cases
finding, sketching and interpreting suitable geometric
representations, finding limits, deciding whether a function is
continuous
Problems:[A]
13.1. 7, 12, 14, 22, 32, 39
10.5. 4, 5, 9, 13, 17, 19
13.2. 3, 5, 7, 9, 13, 15
Lecture 6: part
1 part
2
Content: Differential calculus for
functions of several variables I
Material: [A] 13.3, 5, 6 (in part) , 7 (definition of the
gradient),
Sheets
Concepts: partial derivatives, linear approximation, gradient,
chain rule, directional derivatives
Abilities: Calculating partial derivatives, linearizations,
gradients, directional derivatives, applying the chain rule
Problems:[A]
13.2. 3, 5, 7, 9, 13, 15
13.3. 6, 10, 11, 15, 20, 28, 30
13.5. 3, 9,11, 13, 15,
13.6. 2, 4, 5
Homework Set 3
Lecture 7: part
1 part
2
Concepts: Implicit Function theorem
in 2D, higher order partial derivatives, multiindex notation
Abilities: applying differential calculus to standard geometric
constructions (tangent planes to graphs and level surfaces,
tangent lines to level lines), applying the Implicit Function
Theorem in 2D
Problems:[A]
13.7. 4, 6, 8, 15, 18, 31
13.8. 1,5,7
Homework
Set 4
Lecture 8: part
1 part
2
Concepts: Higher order partial and
directional derivatives, Taylor approximation, Taylor's theorem.
Abilities: Calculating higher order (directional) derivatives, use
of multiindex notation, calculation of Taylor polynomials
Problems:[A]
13.4. 4, 6, 11
13.5. 19, 20
13.8. 22
13.9 3, 4, 5, 8, 9, 15
Content: Optimization I
Material: [A] 14.1, without Theorem 3, including second derivative
test for functions of 2 variables, see Remark p. 750;
14.2 until "Linear Programming"
Concepts: local / global maximum / minimum, critical point, saddle
point
Abilities: finding and (in 2D) classifying of critical points and
extrema on domains with and without boundary
Problems: [A]
14. 1. 1, 4, 8,
14, 16 (three variables: only finding the critical points), 19,
23, 26
14.2. 1, 3, 5, 8, 11
Lecture 10: part
1 part
2
Content: Optimization II
Material: [A] 14.3 until "Nonlinear programming"
Concepts: constraints, Lagrange multiplier, Lagrange equation
Abilities: formulating and solving of Lagrange equations for
differentiable optimization problems with equality constraints
Problems: [A]
14.3 1, 3, 7, 8 (Note: the length of the major and
minor half axis of an ellipse are the maximal and minimal
distances of points on the ellipse to the center), 10, 14, 17,
18, 20
Lecture 11: part
1 part
2
Content: Differential calculus
for functions of several variables IV: vector valued functions
Material: [A] 13.6, 13.8,
Pencast
Concepts: vector valued functions of several variables, their
derivatives and linearizations, Jacobi matrices, chain rule for
vector valued functions, Implicit Function Theorem (general case)
Abilities: Calculation of derivatives and linearizations for
functions of several variables, applications of the Implicit
Function Theorem to systems of equations
Problems:[A]
13.6. 18, 19,
13.8. 13, 14, 16 ,17,18 (for the notation in the last two
problems, see [A] 13.8, "Choosing dependent and independent
variables"
Lecture 12: part
1 part
2
Content: Integral calculus for functions of several
variables I
Material: [A] 15.1, 15.2, 15 .5
Concepts: Integral of a function of several variables, domain of
integration
Abilities: Calculation of integrals of functions of several
variables by iterated integrals, interchanging integration order
Problems: [A]
15.1 17,19,21 ("by inspection" means: without
explicitly calculating integrals, by geometric reasoning)
15.2 5,6,15,16, 22, 25
15.5 5,7,10,15,17,19
Lecture 13: part
1 part
2
Content: Integral calculus for functions of
several variables II
Material: [A] 15.3,4,
Concepts: Improper integrals in several variables, Change of
Variables formula
Abilities: Investigation of convergence and calculation of
improper integrals in several variables, applying change of
variables in multidimensional integrals, in particular, in polar
coordinates
Problems: [A]
15.3 1,5,6,13, 15
15.4 7, 9, 11, 18, 30
Lecture 14: part
1 part
2
Content: Integral calculus for functions
of several variables III
Material: [A] 15.6, 15.7 Pencast
Concepts: cylindrical coordinates, spherical coordinates
Abilities: Calculation of integrals in 3D, in particular
using.cylindrical and spherical coordinates
Problems: [A]
15.6 1, 3, 7, 11, 13, 15, 16
15.7 19, 20, 21
Final test November 2017
Solutions
Final test February
2018
Solutions
Final
test November 2018
Solutions
Final
test January 2019
Solutions
Final test October 2019
Solutions
Final
test January 2020
Solutions
Final
test October 2020
Solutions
Final
test January 2021
Solutions
Final test November 2021
Solutions
Final test January 2022
Solutions
Final
test May 2022
Solutions
Final test November 2022
Solutions
Final
test January 2023
Solutions
Final
test November 2023
Solutions
Final
test January 2024
Solutions