syllabus of the course mathematical …ricci/syllabus_mathmethexpscience_2017-2018.pdfsyllabus of...

SYLLABUS of the course

MATHEMATICAL METHODS FOR EXPERIMENTAL SCIENCE

Bachelor in Computer Science and Engineering,University of Bolzano-Bozen, a.y. 2017-2018

Lecturer: LEONARDO RICCI

(last updated on January 18, 2018)

1

(02/10/2017) 1. Introduction to the course. Vectors (part 1 of 2).

• Vectors (part 1 of 2):

– vectors; n-space; scalars;

– properties of vectors;

– graphical representation;

– scalar product and its properties;

– length or norm of a vector;

– Cauchy-Schwartz inequality;

– triangle inequality (derived from Cauchy-Schwartz inequality).

(02/10/2017) 2. Vectors (part 2 of 2). A summary of single-variabledifferentiation.

• Vectors (part 2 of 2):

– perpendicularity or orthogonality (case on R2 and generalization toRn);

– projection of a vector onto another (case on R2 and generalization toRn);

– angle between two vectors (case on R2 and generalization to R

n);

– unit vectors (versors);

– standard basis versors;

– representation theorem;

– graphical representation.

• A summary of single-variable differentiation:

– definition and graphical interpretation;

– derivative of f(x) = x and f(x) = x2;

– derivative of the product of two functions.

(03/10/2017) E1. Exercise class on vectors, and on single-variabledifferentiation.

2

• Solution of exercises on vectors:

– proof of∣

∣

x+y2

∣

∣ 6

√

x2+y2

2 via Cauchy–Schwartz inequality;

– generalization of the result (the modulus of the average is 6 theroot–mean–square) to an arbitrary number of variables;

– projection of A = (1, 2, 3) along B = (1, 2, 2);

– projection of A = (1, 2, 3, 4) along B = (1, 1, 1, 1);

– expression for cos(x − y) by using the angle formed by two vectors;

– a summary of trigonometric formulas.

• Solution of exercises on single-variable differentiation:

– derivative of f(x) = xn, n ∈ N;

– differentiation of a linear combination of functions;

– exercise,

∗ derivative of f(x) = x3 − 1

2x2 + 2;

– derivative of f(x) = sin(x) and f(x) = cos(x).

(09/10/2017) 3. Functions of several variables: definitions, graphs,level curves. A summary of single-variable differentiation.

• From the functions of a single variable to the functions of several variables:

– definition;

– domain;

– range.

• Examples concerning domains and ranges from the textbook:

– paragraph 12.1, exercises 1, 2, p. 675.

• Graphs (expecially of functions of two variables):

– definition;

– graphical visualization of 2-d case (artist’s view);

– graph as an n-dimensional hypersurface in an n+1 dimensional space.

• Level curves (expecially of functions of two variables):

– definition;

– examples,

3

∗ contour lines, or contours (elevation),

∗ isobars and isotherms;

– extension to the 3-d case (level surfaces).

• Examples of graphs and level curves:

– z =√

1− x2 − y2 (via gnuplot);

– z = xy (via gnuplot);

– z =sin

(√x2+y2

)

√x2+y2

(via gnuplot).


– chain rule;

– example of f(x) = sin(x2).

(10/10/2017) 4. Functions of several variables: limits and continuity.A summary of single-variable differentiation.

• A summary on limits in the single variable (1-d) case:

– summary, by using the example limx→3 x2 = 9;

– other examples,

∗ limx→0x|x| ,

∗ limx→0+√x ,

∗ limx→0sin xx .

• Limits in the n-d case:

– definition;

– existence of the limit for every path chosen to reach the limit point,and independence on the path;

– extension of usual 1-d laws of limits (addition, subtraction, multipli-cation, division, composition) to n-d cases;

– extension of the 1-d squeeze theorem to n-d cases (statement only).

• Examples of limits from the textbook:

– paragraph 12.2, example 3, p. 678, lim(x,y)→(0,0)2xy

x2+y2 ;

– paragraph 12.2, example 4, p. 679, lim(x,y)→(0,0)2x2yx4+y2 ;

– paragraph 12.2, example 5, p. 679, lim(x,y)→(0,0)x2y

x2+y2 .

• Continuity:

4

– summary of the single variable (1-d) case;

– definition;

– extension of usual 1-d laws (sum, difference, product, quotient, com-position) to n-d cases.

• Continuous extension to a point:

– continuous extension to a point in 1-d (ex.gr.: extension of y = sin xx

in 0);

– continuous extension to a point (ex.gr.: extension of z = x2yx2+y2 in 0;

see also paragraph 12.2, example 5, p. 679).


– derivative of a quotient;

– example of sin xx .

(10/10/2017) E2. Exercise class on domains and limits of functionsof several variables, and on single-variable differentiation.

• Solution of selected exercises on domains of functions of several variables,from the textbook:

– exercises 3, 4, 5, p. 675.

• Solution of selected exercises on limits of functions of several variables,from the textbook:

– exercises 4, 5, 13 p. 680.


– derivative of f(x) = exp(x);

– exercises,

∗ derivative of f(x) = tan(x),

∗ derivative of f(x) = e−x2/2.

(16/10/2017) 5. Partial differentiation.

• Partial differentiation:

– definition and notation ( ∂∂xf ,

∂f∂x , fx, f1);

– extension of standard single-variable differentiation laws (sum, dif-ference, etc.) to partial differentiation;

5

– examples of partial derivatives from the textbook,

∗ paragraph 12.3, examples 1-3, p. 682-683;

– partial differentiation of the composition g(x, y) = f [u(x, y), v(x, y)](chain rule);

– partial differentiation of the composition g(x, y) = f [u(x, y)] (chainrule);

– (other than in 1-d case...) existence of partial derivatives does notimply continuity (ex.gr. z defined as 2xy

x2+y2 outside the origin and 0

in the origin; see exercise 36 from the textbook, p. 688);

– (as in 1-d case...) continuity does not imply the existence of the

partial derivatives (ex.gr. z =√

x2 + y2).

• Higher-order partial derivatives:

– definition and notation;

– theorem on equality of mixed partials (statement only);

– examples of partial derivatives from the textbook,

∗ paragraph 12.3, examples 1-2, p. 682-683.

• Introduction to O(x) and o(x) notation.

(17/10/2017) 6. Differentiability and linearization.

• O(x) and o(x) notation.

• Differentiability:

– definition of differentiability;

– differentiability in a point implies continuity in that point;

– existence of partial derivatives, continuity, differentiability;

– theorem: if the partial derivatives of a function are continuous in aneighbourhood of a point then the function is differentiable in thatpoint, and thus also continuous (statement only; ex.gr. z = 2xy

x2+y2

outside the origin and z = 0 in the origin).

• Linearizations (linear approximations):

– 1-d case;

– 2-d case (and, more generally, n-d case);

– differentiability as the possibility of linearizing (linearly approximat-ing);

– example from the textbook,

∗ paragraph 12.6, exercise 5, p. 712.

6

(17/10/2017) E3. Exercise class on partial differentiation and lin-earization, and on single-variable differentiation.

• Solution of selected exercises on partial differentiation from the textbook:

– exercise 7, p. 687;



– exercise 5, p. 692.

• Solution of selected exercises on linearization from the textbook:

– example 1, p. 704;



– inverse function theorem (explained by examples),

∗ case of ln(x),

∗ case of√x,

∗ case of arctan(x) (left as a homework).

(23/10/2017) 7. Gradient, differential, directional derivative.

• Gradient:

– definition;

– linearization expressed in terms of gradient.

• Differential:

– definition;

– linearization expressed in terms of differential.

• Directional derivative:

– definition of directional derivative (or rate of change along a givendirection);

– directional derivative expressed in terms of gradient;

– geometrical interpretation of the gradient.

• Examples from the textbook:

– paragraph 12.6, exercise 1, p. 712;

– paragraph 12.7, example 2, p. 716-717;

– paragraph 12.7, example 3, p. 718;

– paragraph 12.7, example 4, p. 718-719;

7

– paragraph 12.7, exercise 12, p. 723.

(24/10/2017) 8. Extreme values.

• Introduction to extreme values:

– local maxima and minima (from 1-d to n-d);

– examples,

∗ z = x2 + y2,

∗ z =√

1− x2 − y2,

∗ z =√

x2 + y2,

∗ z = xy,

∗ z = x2 − y2.

• Necessary and sufficient conditions for the existence of extreme values:

– definition of critical (stationary) points, singular points, boundarypoints;

– theorem on the necessary conditions (with proof);

– conditions are not sufficient (counterexamples: saddle points like z =xy, z = x2 − y2);

– theorem on the sufficient conditions (without proof).

• Determination of the characteristics of a critical point via Hessian matrix(2-variable case only):

– a summary on the definiteness of a matrix;

– definition of Hessian matrix;

– determination of the characteristics of a critical point out of the def-initeness of the Hessian matrix (without proof);

– in the case of indetermination... (brute force) characterization ofcritical points via analysis of neighbourhoods;

– example, z = 2x3 − 6xy + 3y2 (from the textbook; paragraph 13.1,example 7, p. 748).

(24/10/2017) E4. Exercise class on: gradient, directional derivative;extreme values.

• Solution of a selected exercise on gradient and directional derivative fromthe textbook:

8


• Solution of selected exercises on extreme values (mostly from the text-book):

– examples, z = x4 + y4, z = x3 + y3;

– example 1, p. 751;

– example 2, p. 751-752;


(30/10/2017) 09. Curves. Lagrange multipliers.

• Curves:

– expression by means of functions of the kind g(x, y, ...) = 0;

– displacement on a curve;

– a remarkable application: parabolic reflector.

• The method of Lagrange multipliers:

– heuristic justification of the method;

– general statement (without proof) of the method

(F = f − λg; ~∇F = 0; ∇g 6= 0, no endpoint of the curve);

– remarkable examples from geometry,

∗ shape of the rectangle that can be inscribed in a circle and hasthe largest area,

∗ distance of a point from a straight line.

(31/10/2017) 10. Method of least squares. A summary of single-variable integration.

• Fitting data to a straight line (linear regression), via method of leastsquares:

– merit function given by the sum of the squares of the residuals;

– χ2 merit function in case of uncertainties on y, and evaluation of thestraight line;

– example from the textbook (example 1, p. 767);

– short discussion on related topics,

∗ linear fit,

∗ nonlinear fit and the Levenberg-Marquardt algorithm.

9

• Linear fit with a constant value:

– the weighted mean;

– the sample mean.

• A summary of single-variable integration:

– graphical interpretation and definition;

– definite and indefinite integral (aka primitive aka antiderivative);

– the fundamental theorem of calculus;

– not to be forgotten: the “+c”!

–∫ a

b f(x)dx = −∫ b

a f(x)dx ,∫ b

af(x)dx =

∫ c

af(x)dx+

∫ b

cf(x)dx ,

∫ b

a [αf(x) + βg(x)]dx = α∫ b

a f(x)dx + β∫ b

a g(x)dx ;

– indefinite integral of xn, sin(x), cos(x), ex, 1/x.

(31/10/2017) E5. Exercise class on Lagrange multipliers, and onsingle-variable integration.

• Solution of selected exercises on Lagrange multipliers:

– shape of the box that can be inscribed in a sphere and has the largestvolume;

– from the textbook, exercise 2, p. 764;

– from the textbook, exercise 22, p. 764.

• Solution of exercises on single-variable integration:

– exercise,

∗ indefinite integral of f(x) = x3 − 1

2x2 + 2;

– integration by parts;

– indefinite integral of f(x) = x sin(x), f(x) = x cos(x).

(06/11/2017) 11. Metric spaces and function spaces: an introduction.

• The issue of approximating:

– an example in R3 (and special case of orthonormal vectors);

10

– possibility of extending the method to functions;

– the importance of defining a metric in order to approximate a func-tion by a linear combination of other functions.

• Metric spaces:

– vector space;

– metric (distance) in a space;

– norm and related metric;

– example of Rn, with a mention of L2 (Euclidean) and L1 (“city-block”) norm (and metric).

• A simple function space:

– set of the functions that are continuous and bounded on an interval;

– definition of a norm (and thus a metric) via an integral;

– least-squares approximation of a function f(x) in the interval [0, a]by means of a polynomial p+ qx;

– example f(x) = x2 with a = 1.

(07/11/2017) 12. Least-squares approximation of a function by meansof polynomials. Gram-Schmidt orthonormalization method. A sum-mary of single-variable integration.

• A summary of single-variable integration (and differentiation):

– derivative with respect to a limit of integration of a definite integral;

– De L’Hopital’s rule;

– limit case for a → 0 of the least-squares approximation of a functionf(x) in the interval [0, a] by means of a polynomial px+q (see lecture11.).

• Least-squares approximation of a function f(x) in the interval [a, b] bymeans of a linear combination of n orthonormal functions {φ0(x), . . . , φn(x)}:

– generalization of last lecture’s result: distance between a functionf(x) and a linear combination p0φ0(x) + p1φ1(x);

– definition of a scalar product of two functions and its properties;

– expression of the distance between a function f(x) and a linear com-bination p0φ0(x) + p1φ1(x) by using the scalar product;

– the need of an orthonormal set of functions;

– from the case p0φ0(x) + p1φ1(x) to the general solution,

11

fapprox(x) =∑n

i=0 piφi(x), pi =< φi(x)|f(x) >.

• Gram-Schmidt method for the generation of an orthonormal base of func-tions:

– discussion of the method in R3;

– generation of an orthonormal set of polynomials of degree 0, 1, 2 in[0, 1].

(07/11/2017) E6. Exercise class on approximation of a function bypolynomials, and on single-variable integration.

• Approximation of functions by polynomials:

– normalization of the second-degree base polynomial in [0, 1],x2 − x+ 1

6 ;

– least-squares approximation of the function y = x3 by a second-degree polynomial in [0, 1];

– least-squares approximation of the function ex by a second-degreepolynomial in [0, 1].


– integration by substitution;

– normalization of the second-degree base polynomial in [0, 1],x2 − x+ 1

6 , carried out by substituting x− 12 with y;

– recursive calculation of the integrals∫ 1

0xnex dx.

(13/11/2017) 13. Complementary topics: Taylor and Maclaurin ex-pansion. Fourier series (part 1 of 2).

• A mention of Taylor and Maclaurin expansion:

– general expression;

– examples of expx, cosx, sinx.

• Fourier orthonormal base in [−π, π]:

– scalar product (with normalization 1/π) within the interval [−π, π];

– description of the orthonormal base;

– plot of the first functions.

• Expansion in Fourier series and Fourier theorem:

12

– Fourier series;

– Fourier expansion of a function defined in [−π, π), and Fourier coef-ficients;

– example of unlimited total fluctuations, sin(1/x) in x = 0;

– Fourier theorem (without proof) for a function defined in [−π, π),with integrable modulus and limited total fluctuations;

– case of a continuous function.

• Further aspects of Fourier series:

– periodic extension;

– odd and even functions,

∗ decomposition of a function in an odd and an even part,

∗ Fourier coefficients.

(14/11/2017) 14. Fourier series (part 2 of 2).

• Fourier series for a periodic function with generic period T :

– new scalar product and orthonormal base;

– “angular frequency” ω ≡ 2π/T .

• A remarkable example: Fourier expansion of a square wave (sign(x) =|x|/x).

• Continuity and rate of convergence of the Fourier coefficients:

– discontinuity and continuity of the periodic extension and its deriva-tives (up to the k-th), and n-dependency of the Fourier coefficients(O(1/n), O(1/nk+2));

– example of the square wave.

• Parseval’s theorem for a Fourier series:

– proof of the theorem;

– example of the square wave (and a first evaluation of π, upon adiscussion on the determination of e).

• Integration and differentiation of a Fourier expansion:

– integrability;

– differentiability and related condition on the continuity of the peri-odic extension;

– example of the square wave,

∗ non differentiability,

∗ integration between 0 and x and Fourier expansion of the func-

13

tion |x|.

(14/11/2017) E7. Exercise class on Fourier series, the solution of theBasel problem, and on single-variable integration.

• Fourier expansion of y = x (base period [−π, π)):

– rate of convergence of the Fourier coefficients;

– evaluation of the Fourier coefficients;

– application of Parseval’s theorem and evaluation of∑∞

n=1 1/n2 (Basel

problem).

• Fourier expansion of y = x2 (base period [−π, π)):

– rate of convergence of the Fourier coefficients;

– evaluation of the Fourier coefficients;

– application of Parseval’s theorem and evaluation of∑∞

n=1 1/n4;

– evaluation of the Fourier expansion of y = x via differentiation of theFourier expansion of y = x2;

– mention of the evaluation of the Fourier expansion of y = x2 viaintegration of the Fourier expansion of y = x.


– calculation of the integral∫

eβx cosαxdx;

– calculation of the integral∫

eβx sinαxdx.

(20/11/2017) 15. Multiple integration (part 1 of 3).

• Introduction to double integration:

– heuristic interpretation as a volume;

– definition as a limit (relying on Riemann sums).

• Integration domains:

– simple and regular domains;

– integrability of a bounded, continuous function on a bounded, regulardomain (without proof);

– improper integrals.

• Iterated integrals:

– iteration in the case of simple domains;

14

– examples from the textbook (in all three cases both iterations of asimple domain were used),

∗ paragraph 14.2, example 1, p. 798-799,


∗ exercise 9, p. 802.

(21/11/2017) 16. Fourier series (part 3 of 3). Complex numbers.

• Fourier analysis of the response to a periodic excitation of a “low-passfilter” system, described by the differential equation y + y/τ = f(t)/τ :

– general solution;

– solution in the case of a constant excitation;

– solution in the case of a square wave excitation.

• Complex numbers:

– definition,

∗ any real number is a complex mumber,

∗ a complex mumber i exists | i2 = −1,

∗ any complex number z can be written as z = x+iy, with x, y ∈ R

(x, y being the real and the imaginary part of z, respectively),

∗ the ordinary arithmetic properties of addition and multiplicationare conserved;

– set C of the complex numbers;

– graphical representation (Argand-Gauss plane);

– complex conjugate,

∗ definition of z, z ≡ x− iy,

∗ graphical representation,

∗ evaluation of the real and the imaginary part of a complex num-ber, Re z = x = z+z

2 , Im z = y = z−z2i ,

∗ complex conjugate of addition, subtraction, multiplication, z ± v =z ± v, z · v = z · v;

– modulus,

∗ definition |z|, |z| ≡√zz =

√

x2 + y2,

∗ graphical representation,

∗ modulus of the complex conjugate, |z| = |z|,∗ modulus of multiplication, |z · w| = |z| · |w|;

15

– division of two complex numbers,

∗ via solution of a system,

∗ v/z = vz/|z|2,∗ example: calculation of (2 + 3i)/(4− 2i);

– Euler’s formula,

∗ derivation from Maclaurin expansions of ex, cos(x), sin(x),

∗ example: calculation of√i (via system of two variables x and y,

and via Euler’s formula),

∗ representation of a complex number z as |z| eiθ, with θ = arctan(Re z/Imz).

(21/11/2017) E8. Exercise class on multiple integration via iteratedintegrals.

• Solution of selected exercises on double integration from the textbook:

– example 3, p. 800 (both iterations were, at least tentatively, used);

– exercise 15, p. 802 (both iterations were, at least tentatively, used);

– example 3, p. 804 (improper integral due to unbounded function);

– example 1, p. 803 (improper integral due to unbounded domain);

– exercise 2, p. 807 (improper integral due to unbounded domain).

• Volume of a right pyramid with a square base.


• Area and volume evaluation:

– area evaluationvia

∫∫

DdA · 1;

– evaluation of the surface area of a circle;

– volume evaluation as an integral on a surface f(x, y)via

∫∫

DdA · f(x, y);

– evaluation of the volume of a sphere;

– extension of the concepts of double integration to the case of morethan two variables;

– volume evaluation as a volume integral∫∫∫

DdV · 1.

• From Cartesian coordinates to polar, cylindrical and spherical ones:

16

– polar coordinates,

∗ transformation and inverse transformation,

∗ area element;

– cylindrical coordinates,


∗ volume element;

– spherical coordinates,


∗ volume element,

∗ area element at fixed radius;

– evaluation of the surface area of a circlevia

∫∫

DdA · 1 and polar coordinates;

– evaluation of the volume of a spherevia volume integral

∫∫∫

D dV · 1 and spherical coordinates;

– evaluation of the surface area of a sphere via spherical coordinates.


• Evaluation of the Gaussian integral.

• Coordinate transformation (mapping) and Jacobian determinant:

– Jacobian determinant;

– Jacobian determinant of the inverse transformation;

– trasformation from Cartesian to polar coordinates and from polarcoordinates to Cartesian,

∗ evaluation of ∂(x, y)∂(ρ, θ) ,

∗ evaluation of ∂(ρ, θ)∂(x, y) ,

∗∣

∣

∣

∂(x, y)∂(ρ, θ)

∣

∣

∣= 1/

∣

∣

∣

∂(ρ, θ)∂(x, y)

∣

∣

∣.

• Surface area of an elliptical disk.

• Example from the textbook (paragraph 14.4, example 8, p. 815).

(28/11/2017) E9. Exercise class on change of variables in doubleintegration.

• Solution of selected exercises on change of variables in double integrationfrom the textbook:

17




– exercise 5, p. 817, with y2 instead of x2;



(04/12/2017) 19. A classification of differential equations. Boundaryconditions. Linear, scalar, ordinary differential equations.

• Definition of differential equations (DE ).

• A classification of DE:

– ordinary and partial DE;

– scalar and vectorial DE;

– order of a DE;

– The further discussion is restricted to scalar, ordinary DEs (scalarODEs)F (x, y′, y′′, . . . y(n)) = 0;

– linear and nonlinear DE;

– homogeneous and nonhomogeneous DE.

• Order of a DE and boundary conditions:

– number of unknown constants to be expected when solving a DE, i.e.number of boundary (initial) conditions required, equal to the orderof the DE (without proof);

– Cauchy problem.

• Linear, scalar ODEs of any order:

– expression,∑n

k=0 ak(x)y(i)(x) = f(x);

– homogeneous case, and linear combination of two solutions;

– nonhomogeneous case, and sum of a solution of this case with asolution of the homogeneous case;

– solution of linear, constant-coefficients, homogeneous, scalar

18

ODEs via auxiliary equation (without proof);

– examples from the textbook:

∗ paragraph 17.5, examples 1a, 1b, p. 959.

(05/12/2017) 20. First-order scalar, ordinary differential equations.

• First-order, separable, scalar ODEs:

– description (DE can be nonlinear) and solution;

– examples from the textbook,

∗ paragraph 7.9, animal population growth example,

∗ paragraph 7.9, example 1, p. 446,

∗ paragraph 7.9, example 2, p. 446.

• First-order linear, homogeneous, scalar ODEs:

– description and solution as a separable DE;

– solution in the case of constant coefficients, also by means of theauxiliary equation;

– a remarkable example,

∗ dydt +

yτ = 0,

∗ decay time and half-life.

• First-order linear, nonhomogeneous, scalar ODEs:

– nonhomogeneous case solved by using the solution of the homoge-neous case;

– examples from the textbook,


∗ paragraph 7.9, example 8, p. 450.

(05/12/2017) E10. Exercise class on first-order scalar, ordinary dif-ferential equations.

• Solution of selected exercises on first-order scalar ODEs from the textbook:







19

• Partial fraction decomposition.

(11/12/2017) 21. Second-order linear, scalar, ordinary differentialequations with constant coefficients.

• Second-order linear, constant-coefficients, homogeneous, scalarODEs:

– solution via auxiliary equation;

– example from the textbook (paragraph 3.7, example 3, p. 205);

– oscillations;

– example from the textbook (paragraph 3.7, example 5, p. 207).

• Solution of second-order linear, constant-coefficients, nonhomo-geneous, scalar ODEs via method of “variation of constants” andWronskian determinant:

– method of “variation of constants”, Wronskian determinant;

– example from the textbook (paragraph 17.6, example 1, p. 962).

• Solution of second-order linear, constant-coefficients, nonhomo-geneous, scalar ODEs via method of “undetermined coefficients”:

– discussion in the case of a polynomial nonhomogeneous term,

∗ example from the textbook (paragraph 17.6, example 1, p. 962;see above),

∗ example above with x2 instead of 4x as nonhomogeneous term;

– summary of the method in the case of polynomial, exponential, andsinusoidal nonhomogeneous terms (also when these terms are propor-tional to a solution of the associated homogeneous equation) [*],

∗ example above with ex instead of 4x as nonhomogeneous term.

[*] discussed in lecture 22.

(12/12/2017) 22. Linearity and solution of differential equations.Numerical methods for the solution of ordinary differential equations.

• Exploitation of linearity in the solution of second-order linear,constant-coefficients, nonhomogeneous, scalar ODEs where thenonhomogeneous term is given by a linear combination of differ-ent terms:

– example from the textbook (paragraph 17.6, example 1, p. 962) with4x2 − 4x instead of 4x as nonhomogeneous term.

20

• Euler method (first-order Runge-Kutta method) for the numerical inte-gration of first-order scalar ODEs.

• Runge-Kutta method for the numerical integration of first-order scalarODEs:

– discussion of second-order Runge-Kutta method;

– mention of fourth- and higher-order Runge-Kutta methods, as well asof the availability of numerical libraries (ex.gr. GSL – Gnu ScientificLibrary).

• Generalization of Runge-Kutta methods from the scalar to the vectorialcase.

• Reduction of an n-th order vectorial (or scalar) ODE to a first-order vec-torial one:

– discussion by using the numerical solution of the two-body gravita-tional interaction (ex.gr. orbit of the earth around the sun) as anexample.

(12/12/2017) E11. Exercise class on second-order linear, scalar, ordi-nary differential equations with constant coefficients.

• Solution of selected exercises on second-order linear, constant-coefficients,scalar ODEs from the textbook:

– examples 2a, 2b, 2c, p. 962-963, exercises 11, 12, p. 967, solved via

∗ method of “variation of constants” and Wronskian determinant(examples 2a, 2b, each single term of exercise 11),

∗ method of “undetermined coefficients” (examples 2a, 2b, eachsingle term of exercise 11, exercise 12),

∗ exploitation of linearity (example 2c, exercise 11).

(18/12/2017) 23. Summary of the course (part 1 of 2).

• Summary of differential calculus of functions of two or more variables:

– differentiability and related theorem (statement only: if partial deriva-tives of a function exist in a neighbourhood of a point and are contin-uous in that point, then the function is differentiable in that point);

– gradient, differential, directional derivative;

– linearization / linear approximation;

– extreme values (critical, singular, boundary points), Hessian matrix

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H =

∂2z∂x2

∂2z∂x∂y · · ·

∂2z∂y∂x

∂2z∂y2 · · ·

......

. . .

;

– Lagrange multipliers;

– examples from past exams:

∗ exercise 1 of 06.02.2013 exam,

∗ exercise 2 of 18.06.2013Mathematical Methods for Physics exam.

• Summary of integral calculus of functions of two or more variables:

– multiple integration via iterated integrals;

– change of variables in double integration, Jacobian determinant, Ja-cobian determinant of the inverse transformation

∂(x,y,... )∂(u,v,... ) = det

∂x∂u

∂x∂v · · ·

∂y∂u

∂y∂v · · ·

......

. . .

;

– examples from past exams:


∗ exercise 5 of 06.02.2013 exam (with an additional variation, i.e.the integration of x2 + y on the unitary circle.

(19/12/2017) 24. Summary of the course (part 2 of 2).

• Summary of integration of scalar ODEs:

– Cauchy problems;

– linear, constant-coefficients, homogeneous, scalar ODEs (solution viaauxiliary equation);

– first-order scalar ODEs,

∗ separable DE,

∗ linear, homogeneous DE,

∗ linear, nonhomogeneous DE;

– second-order linear, constant-coefficients, scalar ODEs,

∗ homogeneous DE,

∗ solution of nonhomogeneous DE via method of “variation of con-stants” and Wronskian determinant,

∗ solution of nonhomogeneous DE via method of “undeterminedcoefficients”;

∗ exploitation of linearity;

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– examples, mainly from past exams:

∗ y′′′ − 3y′′ + 3y′ − y = 0,


∗ from the textbook, paragraph 7.9, example 8, p. 450,

∗ y′′ − 2y′ + y = tet,

∗ exercise 3 of 06.02.2013 exam (with an additional variation, i.e.the integration of the DE with the nonhomogeneous term givenby αe2x + β sinx).

(19/12/2017) E12. Summary exercises.

• Solution of selected exercises from past exams:

– (morning session)

∗ exercise 1e of 06.02.2013Mathematical Methods for Physics exam,

∗ exercises 1-5 of 03.07.2017 exam;

– (afternoon session)

∗ exercise 1e of 18.06.2013Mathematical Methods for Physics exam,

∗ exercises 1-5 of 08.09.2017 exam.

(18/01/2018) — Supplementary exercise class.Duration: 2h30’

• Supplementary exercise class: solution of selected exercises:

– exercises 1-5 of 08.02.2016 exam.

(25/01/2018) — Supplementary exercise class.Duration: 2h30’

• Supplementary exercise class: solution of selected exercises:

– exercises 1-5 of 29.06.2016 exam.

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syllabus of the course mathematical …ricci/syllabus_mathmethexpscience_2017-2018.pdfsyllabus of...

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