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.
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.
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 thefirst 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.
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.
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-calledtranslation, 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 translationi.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.
the integral of a linear combination is the linear combination of the integrals,
If a > b then define
Additivity of integration on intervals. If c is any element of [a, b], then
Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded on that interval. Thus there are real numbersm and M so that m ≤ f (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(b − a) and
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’.