Kategorier
Calculus matematik 5

Numerical methods for calculation of integrals

For many functions it is impossible to find a primtive function and therefore it is impossible to use the fundamental theorem of calculus to solve the integral.

Luckily there are ways to cope with such circumstances with the aid of numerical methods. Here one has deviced a formula for approximation of the integral.

The perhaps the simplest one is the trapezoidal formula.

The principle is to divide the area into well-known entities( in this case trapezoids) for which the area may readily be computed.

A more refined method is thee.

 Simpson’s rule. This can be viewed as a combination of the midpoint-rule and the trapezoidrule. The midpoint rule basically amounts to approximating the area by rectangles. 

Kategorier
Advanced

Vector algebra-adding and subtacting vectors

Garden

The mass of your body is a measure of the amount of matter (atoms) that constitute your body. This number is a quantity that can be pinpointed on the x-axis (or any line of numbers). Such quantities are called scalars

It is important here to bear in mind the difference between weight and mass.

The weight is the pull of gravity acting upon any mass within the gravitational field.  This quantity not only has a size but also a direction. It is directed towards the center of gravity of the earth.

Galaxy M 83 15 million light-years away held together by gravity. Courtesy: Anglo-Australian telescope.

Such quantities possessing both mass and direction are termed vectors.

Many important physical quantities are not just quantities (As eg mass, energy and temperature) but  they also have a direction (as e.g. velocities, forces, momentum, pressure, accelerations).

They are usually represented by the length and direction of an arrow or by the coordinates (x, y, z) of the endpoint of the arrow representing the vector beginning at the origin. They are often denoted by a bold letter (F) with an arrow above it.

Position vector

Vectors are added by the polygonal method which means that in order to obtain the vectorsum of several vectors you let the vector number two start at the endpoint of the first vector and so on. After you have drawn all the vectors like this after each other you are able to construct the vectorsum, or resultant, of all the vectors by drawing one vector from the startpoint of the first vector to the endpoint of the last vector.

File:Vectoraddition.svg

A vector can always be divided into x,y, and z-components

F=Fx i+ Fy j+ Fz k.

where i, j, k are the orthogonal unit vectors for the cartesian coordinatesystem.

This gives us the possibility to add vectors algebraically:

 F + G =  (Fx+Gx, Fy+Gy, Fz+Gz) i.e. add the x-coordinates separately and do likewise with the y and z-coordinates to acquire the coordinates of the sum.

Kategorier
Calculus Gymnasiematematik(high school math) Uncategorized

Symmetries, translations and dilatations

A good definition of symmetry was given by the mathematician Hermann Weyl: 
An object is symmetrical  if, after you performed an operation on it, it still looks the same as it did before.
  • A function can be moved in the horizontal direction so-called translation, by adding or subtracting a number. For example :

f(x) = sin(x-a) is the function f(x) = sin(x) moved a steps to the right.

  • The graph of a function can be moved in vertical direction (vertical translation) by adding or subtracing a number to the function e.g. f(x) = sin(x) +4 is the function f(x) = sin(x) moved four steps upwa
  •  A vectorial translation i.e. a translation in both the x- and the y- direction.  y= sin (x+a) +b.
  • By dilatation is meant a contraction or comprimation of the function. f(x) = sin(x/m) is an extraction of the function f(x)=sin(x/m) by the scale m.
  • A function may also be mirrored in the x-axis. -f(x) is the mirror -image of f(x) in the x-axis. f(x)=-sin(x) is the mirror-image of f(x) = sin(x) in the x-axis.
  • f(-x) is the mirror-image in the y-axis of f(x). f(x)=sin(-x) mirrors f(x) = sin(x) in the y-axis.
  • By combining both types of mirroring one obtains a mirroring in the origin of the function.
Kategorier
Calculus matematik 3c matematik 4

Properties of integrals

  1.  the integral of a linear combination is the linear combination of the integrals,
     \int_a^b (\alpha f + \beta g)(x) \, dx = \alpha \int_a^b f(x) \,dx + \beta \int_a^b g(x) \, dx. \,

If a > b then define

\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx.

3.

Additivity of integration on intervals. If c is any element of [a, b], then \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx.
4.
Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded on that interval. Thus there are real numbers m and M so that mf (x) ≤ M for all x in [a, b]. Since the lower and upper sums of f over [a, b] are therefore bounded by, respectively, m(ba) and
M(ba), it follows that
      m(b - a) \leq \int_a^b f(x) \, dx \leq M(b - a).
 

20140507-183437.jpg

Applications in physics/technology of integrals.

Kategorier
Calculus Gymnasiematematik(high school math) matematik 3c matematik 4

Definite integrals

In a geometrical context the integral can be interpreted as the area between the graph of the integrand f(x) and the x-axis. The idea is to summarize infinitely many infinitely thin rectangles. (An alternative approach to areacomputation is the Lebesgue-integration where you partition the range of f(x)  instead of the domain). This method works better for integration of infinite series e.g. Fourier series than the Riemann integral.

The integral sign is a stylized version of the letter ‘s’.

Bernhard Riemann discovered that this can be accomplished by forming one ‘uppersum’ and one ‘lowersum’.

An undersum can be seen here and to see the oversum click here.

Riemann showed that there exists exacctly one number which is smaller than the uppersum and bigger than the lower sum in the limit when the width of the rectangles approaches zero.

This number is identical to the integral over this interval.

Earlier  in the 18th. century Newton and Leibniz discovered the fundamental theorem of calculus. This gives an easy method for computation of the integral with the aid of the primitive function

F'(x) =f(x)  then       F(x) = \int_a^x f(t)\, dt\,.

See the derivation of this theorem here in an excerpt from  the Gamma textbook (björup et. al.) Fundamentaltheorem.

F'(x) = (F(x+dx) -F(x))/dx   F’ (x) dx = (F(x+h) – F(x)).

By Summation of  all these differential areaelements   it can be seen that all terms cancel each other except the first and the last which gives us the fundamental theorem of calculus:

F(b) - F(a) = \int_a^b f(x)\,dx,


pp.175-180

Ex
=
∫x2dx = x3/3 = 33/3 -0 /3= 27/3 = 9.