化學數學（一）. The Mathematics for Chemists (I) (Fall Term, 2004) (Fall Term, 2005) (Fall Term, 2006) Department of Chemistry National Sun Yat-sen University. Chapter 1 Review of Calculus. Numbers and variables Units Algebraic, transcendental, complex functions Coordinate systems Limit
The Mathematics for Chemists (I) (Fall Term, 2004)(Fall Term, 2005)(Fall Term, 2006)Department of ChemistryNational Sun Yat-sen University
Assignment for Chapter 1 :
pp. 142-144: 18,37,45,51,66,71
p.241: 19, 28
Integers (natural, whole, positive, negative, even, odd, composite, prime)
irrational (surds, transcendental))
fixed point and floating point
The discover (Hippasus) of first irrational number
was thrown into sea.
Table: Basic quantities for the atomic unit system
Table: Quantities for the atomic unit system
a+b=b+a, ab=ba (commutative)
a+(b+c)=(a+b)+c, (ab)c=a(bc), (associative)
Roots (zeros of f(x))
(Here the roots of P(x) are the singularities of f(x))
(Proof) (classroom exercise)
Recall the properties of
Why is mathematical induction valid and exact?
An equation is worth infinite number of data; a proof infinite number of examples.
Prove any of above sums
Necessary for convergence:
Further test of convergence:
d’Alembert’s ratio test:
xDifferentiation as Limit of Division
What is dx?
It is a variable. It can be as small as required. Its limit is zero, but it is
absolutely not the same as zero.
The existence of dx relies on a great property (continuity) of real numbers.
The discovery of infinitesimal is one of the greatest discoveries in science.
How about odd n? (Classroom exercise)
Local minima: E,C
Global minimum: C
Local maxima: A, D, F
Global maximum: D
PSnell’s Law of Refraction
To find point O so that the time used for
the light to travel from P to Q is minimized.
(Principle of least time)
Verify the expression for the most probable speed.)
Find the maximum of species I.
The fundamental theorem of the calculus:
The definite and indefinite integrals.
xAverage of a Function
cSpecial Case: Improper Integrals
Calculate the mean speed of N2 at 25 oC
xFunctions of Several Variables
xChange of Variables(Coordinate Transform)
xCurvilinear Integrals Independent of Path
It depends on the initial and final coordinates only.
TEntropy: A State Function
Which path is easier for us to calculate?
Finish the last step.
dxdy = |J|rdrdθ
xFunctions in 3 Dimensions
xGeneral Curvilinear Coordinates
Cylindrical polar coordinates:
Classroom exercise: write the volume element
In cylindrical polar coordinates.