Mathematical metods for physical chemistry (N403045)

Organization

The first part of this course is taught by doc. Malijevský. This part is closed by a written test (T1).
Second part of this course is taught by prof. Kolafa. It includes homework (HW) and is closed by a written test (T2).
Final grade is determined by the weighted average:
grade = T1*0.5 + T2*0.4 + HW*0.1
where 50–60 = E, ... 90–100 = A.
For homeworks, you may use a mathematical assistant (e.g., Maple) or a pen and paper. Web-based tools may help; e.g., Symbolab.
Use extension .mw for Maple files; if you like .mws, use only ASCII characters (no Greek, no Czech).
Every day over the deadline is penalized by 1 point.
Please send your homework by e-mail with file name derived from your name and the homework number (e.g., Novak-1.mw). If you use pen and paper, please scan your homework (a photo by a mobile phone is enough) and e-mail.
Deadline for homework is 10 days after the lecture covering the topic

# lecture print exercises (Maple) homework deadline

1 Linear algebra 1 1:8 mmpc1.mw Consider a plane and a line given by equation ax + by = 0. Write a 2×2 matrix, M, of the mirror reflection (flip) around this line. Calculate det(M) and M² and discuss the results.

2 Linear algebra 2 1:8 mmpc2.mw Determine the signature of quadratic form f(x,y,z)=2*(x-y)^2+(y-z)^2-2*x*z.

3 Approximation of functions 1:8 mmpc3.mw Consider function
f(x) = cosh(√x) for x≥0
f(x) = cos(√−x) for x<0

Show that it is analytical at x=0 and find its Taylor series.

Aproximate this function in interval [−π², π²] by the Chebyshev polynomials with the precision better than 10⁻⁷.

4 Numerical derivative and quadrature, ODE 1:8 mmpc4.mw Derive a fourth-order formula for the numerical calculation of the first derivative in the form
f '(x) = [ (a f(x−3h/2) + b f(x−h/2) + c f(x+h/2) + d f(x+3h/2) ]/h + O(h⁴)
Compare with the standard formula using a suitable test function.

5 Mathematical statistics 1:8 mmpc5.mw Student's t-test: Your personalized homework
6 The method of least squares 1:8 mmpc6.mw Fitting in physical chemistry: Your personalized homework

7 + popular appendix 1:8

#	lecture	print	exercises (Maple)	homework	deadline
1	Linear algebra 1	1:8	mmpc1.mw	Consider a plane and a line given by equation ax + by = 0. Write a 2×2 matrix, M, of the mirror reflection (flip) around this line. Calculate det(M) and M² and discuss the results.
2	Linear algebra 2	1:8	mmpc2.mw	Determine the signature of quadratic form f(x,y,z)=2(x-y)^2+(y-z)^2-2x*z.
3	Approximation of functions	1:8	mmpc3.mw	Consider function f(x) = cosh(√x) for x≥0 f(x) = cos(√−x) for x<0 Show that it is analytical at x=0 and find its Taylor series. Aproximate this function in interval [−π², π²] by the Chebyshev polynomials with the precision better than 10⁻⁷.
4	Numerical derivative and quadrature, ODE	1:8	mmpc4.mw	Derive a fourth-order formula for the numerical calculation of the first derivative in the form f '(x) = [ (a f(x−3h/2) + b f(x−h/2) + c f(x+h/2) + d f(x+3h/2) ]/h + O(h⁴) Compare with the standard formula using a suitable test function.
5	Mathematical statistics	1:8	mmpc5.mw	Student's t-test: Your personalized homework
6	The method of least squares	1:8	mmpc6.mw	Fitting in physical chemistry: Your personalized homework
7	+ popular appendix	1:8

Student's t-test and linar fitting in spreadsheets: [ LibreOffice | Excel ]