Andrea Schafferhans-Fuhrmann I owe the correction of a detail important to the users of the plotter. Paditz for critical hints and suggestions for changes to the bounds of the arguments for complex numbers.
The Empirical Method
Physical Quantities
Units
Ancient Units: Some examples of units still in common use despite the SI convention:. How many laps must Maria and Lucas run until they move back a mile?
Order of Magnitude
The diameter of the range within which scattered electrons sense a proton is about 1.4 fm, atomic nuclei are between 3 and 20 fm thick. Insert: Billion: Although these prefixes of the SI system are internationally established, this is certainly not the case with our well-known number words.
Numbers
- Natural Numbers
- Integers
- Rational Numbers
- Real Numbers
Insert: Powers: Repeated application of the same factor is usually described as a power with the number of factors as . With the real numbers, which have the same rules of calculation of a field as the rational numbers, both solutions of the general.
Boundedness
Monotony
Convergence
For many sequences we can recognize the convergence or even the limit value with some skill just by looking at it. We easily check the theorem of Bolzano and Weierstrass: the sequence (F3)(n1)n∈N is monotonic decreasing and bounded below: 0< 1n, therefore it converges.
Series
The exponential series is thus a majorant for the also monotonically increasing exponential series and ensures the convergence of the series by that of the series. However, when you calculate the members of the series and compare them to the partial sums of the series, you will realize that the series converges much more slowly than the series.
The Function as Input-Output Relation or Mapping
Finally, it is the first step towards a theory, and with it towards an understanding of the experiment. The pre-image set Df is in most cases, like the image set Wf, a part of the axis of real numbers R1.
Basic Set of Functions
- Rational Functions
- Trigonometric Functions
- Exponential Functions
- Functions with Kinks and Cracks
In the following figure, a unit circle is rotated rotatably around the center and carries a virtual ink pattern on its circumference at the end of the red radius. Using the projection of the rotary pointer, the pattern has drawn for us on the 2-axis the "length of the opposite leg" in the right triangle constructed by the circular radius of length one axis hypotenuse, i.e.
Nested Functions
Addition, subtraction and division as usual: x+r, x−r, x/r Multiplying with the star instead of the dot symbol: r?x:=r·x, raising to a power with the hat: x∧r :=xr and r∧x:=rx,. If you are ready with the preparations, you should start drawing with the back button.
Mirror Symmetry
For example, c is the even part of the straight line function y = s·x+c and s·x is the odd part.
Boundedness
The standard parabola y = x2 for example and the absolute value function y = |x| is bounded under throughA= 0, and the step function θ(x) is bounded over throughB = 1 and bounded under throughA= 0.
Monotony
Bi-uniqueness
In this case the equation y =f(x) has for all y at most one solution x, and each line parallel to axis 1 hits the graph of the function at most once. 2) Then in a second step they consider the area containing the image elements (which was not very interesting for us) and investigate whether it consists only of the image points or contains other additional points. Thus, in a bijective map the equation y =f(x) has exactly one solution x and the function is invertible.
Inverse Functions
Roots
The example of the parabola has already shown us that the inverse functions of the powers y = xn with integer exponents n ∈ Z are the root functions with fractions as exponents: x=y1n, rewritten as: y= √n. As inverse functions of the polynomials, we get more complicated roots: for example, from y =x2+ 1 which is bi-unique for x≥0, we find x=p.
Cyclometric Functions
Here you need to be very careful with the calculations: especially before using the calculator, you need to familiarize yourself with the domains of inverse functions. The last notation in particular is sometimes confusing, as it can easily be confused with the inverse sine function (sinx)−1= sin1x.
Logarithms
Insert: Notations: Also in this case there are notation problems as with the inverse functions of the trigonometric functions: arsinh x = arcsinh x = arsh x = sinh−1x, etc. It can be shown that y is the square measure of the area of the hyperbola sector colored in Figure 4-11 (between the origin, the moving point, the vertex, and the moving point reflected on the 1-axis), as the 1-coordinate of the moving point (meaning the hyperbolic cosine is equal to x).
Limits
We therefore say that the exponential function increases more strongly than any power function. We consider the sector A0b of the unit circle with the central angle x near 0, the line segments: |0A|= 1 and |0b|= 1 and the arc(Ab) above the angle x, as well as the point a on the line segment |0A|, the line segments |0a|= cosx and|ab|= sinx line segment, and the |0b extension of| to B, so that the line segment is.
Continuity
What simple hypothesis about the functional dependence of the measured quantity y = f(x) on the varied quantity x would you set up. From the figure we recognize that the difference quotient gives the gradient of the secant, which connects the two points (t1, x(t1)) and (t2, x(t2)).
Differential Quotient
Therefore, we get the best linear approximation to the graph of the function f(x) in the vicinity of the point x0, if we choose a straight line with the differential quotient as gradient, and this is of course exactly the definition of the tangent line. If we can ignore this remainder, we find for the real increase of the function ∆f(x).
Differentiability
Thus we obtained an equation in which the symbols df and dx, which in Leibniz's form of the differential quotient were defined only as a quotient, now appear as singles, as "linear parts of increase" and are also defined as non-infinitesimal quantities. The reverse is not true: every continuous function is not differentiable, as we saw in the above example of the absolute value function f(x) =|x|.
Higher Derivatives
The disappearance of the first derivative f0(x0) = 0 at a point x0, the criterion for a horizontal tangent at this point, is a necessary condition for the existence of a local extremum. Sufficient conditions for a local maximum or minimum can only be obtained by looking at the second derivative: f00(x0)>0 means a turn to the left, i.e.
The Technique of Differentiation
Four Examples
This is the characteristic property of the exponential function and the deeper reason for its unique importance in science, that it is identical to its own differential quotient. From these examples we now obtain all desired differential quotients for all functions in our basic set and beyond using the following rules.
Simple Differentiation Rules: Basic Set of Functions
Here we see the differential quotient of a linear combination of functions is equal to the linear combination of the differential quotients. This is the way we can determine the differential quotients of all rational functions R(x) = Pn(x).
Chain and Inverse Function Rules
The following example illustrates the advantages of the Leibniz notation: We are looking for the first derivative of ((x+ 1/x)4−1)3 forx6= 0 :. dwdz)(dzdy)(dydx) according to the chain rule,. We conclude this list of differential quotients, which is also important for the following chapters, with the cyclometric functions and the area functions:.
Numerical Differentiation
Preview of Differential Equations
Later, when dealing with functions of several variables, you will encounter even more complex differential operations: with the help of so-called "partial" derivatives, you will form the gradients of scalar fields and the divergence or rotation of vector fields. Here we want to examine one of the main applications of the differential calculus, and in some detail.
Power Series
We will talk about Taylor series, which allow us to represent and calculate a large number of functions f(x) needed in physics, near a value x0 of the independent variablex, in the form of a power series.
Geometric Series as Model
Form and Non-ambiguity
Our calculations also show us that the function to be represented must necessarily be infinitely many times differentiable for the Taylor series to exist. Although this function is infinitely differentiable, all its derivatives f(n)(0) = 0 vanish at pointx=0, so no Taylor series around 0 can be constructed.
Examples from the Basic Set of Functions
- Rational Functions
- Trigonometric Functions
- Exponential Functions
- Further Taylor Series
In the following table, we have compiled for you the first two terms of the Taylor series for some common functions, to make memorization easier. Surprisingly, it is exactly the Taylor series of the trigonometric sine, but without the change of signs, which sheds light on the nomenclature.
Convergence Radius
Accurate Rules for Inaccurate Calculations
We would also have obtained the same expression if we had calculated the Taylor series for the product function F(x) :=f(x)g(x). We also find the same expression clear if we decompose the Taylor series for the inverse function after the second term.
Quality of Convergence: the Remainder Term
Taylor Series around an Arbitrary Point
The work done is proportional to the force required and proportional to the distance traveled. The advanced question of the work done by a force varying along an arbitrary function is thus brought back to the mathematical problem of determining the area of a rectangle where one side (in this case the top) is replaced by a curve.
Area under a Function over an Interval
For simplicity, we divide the interval of length bbym−1equidistant intermediate points xn = nb/m into m intervals of the uniform length ∆xn =b/m. We have omitted the necessary investigation into whether the result is independent of the choice of interval dissection and the choice of nodes.
Properties of the Riemann Integral
- Linearity
- Interval Addition
- Inequalities
- Mean Value Theorem of the Integral Calculus
The area "above" the function −f(x) (which runs in the fourth quadrant) automatically receives a negative sign in the integral. This definition is in no way inconsistent with our restriction procedure which introduces the integral through the Riemann sum, since in the case of interchanged integration all ∆xn and consequently all dx become negative.
Fundamental Theorem of Differential and Integral Calculus
- Indefinite Integral
- Differentiation with Respect to the Upper Border
- Integration of a Differential Quotient
- Primitive Function
The procedure is completely analogous to the expansion step of the differential calculus from the gradient f0(x0) of a function f(x) at a certain point x0 to the first derivative f0(x) as a function of the variable x. We again replace the constant upper limit b of the definite integral through available y and as above we get.
The Art of Integration
- Differentiation Table Backwards
- Linear Decomposition
- Substitution
- Partial Integration
- Further Integration Tricks
- Integral Functions
- Numerical Integration
At any time, the integrand can be decomposed into the product f0(x)·g(x), so that for one of the factors, e.g. Because of the remaining integral with a characteristic minus sign on the right-hand side, this is not a complete solution to our problem.
Improper Integrals
- Infinite Integration Interval
- Unbounded Integrand
- Motivation
- Imaginary Unit
- Definition of complex numbers
- Gauss Number Plane
- Euler’s Formula
- Complex Conjugation
The real numbersR: z =x are a subset of the set of complex numbers: R ⊂C, that is, all those with Imz =y = 0. Here we want to illustrate a much-used transformation of complex numbers in the Gaussian number plane: The complex relation assigns every complex number z to its often-used "complex" complex by reversing the signs. of the star a line just above the symbol, which is not available to us here).
Calculation Rules of Complex Numbers
Abelian Group of Addition
Thereby, subtraction of complex numbers is possible, just as you are familiar with from the real numbers: Difference−bis the unique solution of the equation z+b =a. It is apparently impossible to determine for two complex numbers which of them is the larger.
Abelian Group of Multiplication
The laws of the abelian group of multiplication are once again derived simply from the corresponding relations for the real numbers. The star can be drawn in parentheses as in the case of the sum.
Functions of a Complex Variable
- Definition
- Limits and Continuity
- Graphic Illustration
- Powers
- Exponential Function
- Trigonometric Functions
- Roots
- Logarithms
- General Power
- Three-dimensional Real Space
- Coordinate Systems
- Euclidean Space
- Transformations of the Coordinate System
The following figure can help you make a mental image of the effect of the function. To demonstrate the influence of the changes in sign, we finally showed you.
Vectors as Displacements
Displacements
Vectors
Transformations of the Coordinate Systems
Addition of Vectors
Vector Sum
Commutative Law
Associative Law
Zero-vector
Negatives and Subtraction
Multiplication with Real Numbers, Basis Vectors
Multiple of a Vector
Laws
Vector Space
Linear Dependence, Basis Vectors
Unit Vectors
Scalar Product and the Kronecker Symbol
Motivation
Definition
Commutative Law
No Associative Law
Homogeneity
Distributive Law
Basis Vectors
Kronecker Symbol
Component Representation
Transverse Part
No Inverse
Vector Product and the Levi-Civita Symbol
Motivation
Definition
Anticommutative
Homogeneity
Distributive Law
With Transverse Parts
Basis Vectors
Levi-Civita Symbol
Component Representation
No Inverse
No Associative Law
Multiple Products
Triple Product
Nested Vector Product
Scalar Product of Two Vector Products
Vector Product of Two Vector Products
Transformation Properties of the Products
Orthonormal Right-handed Bases
Group of the Orthogonal Matrices
Subgroup of Rotations
Transformation of the Products
