- 1 Euclids "common notions"
- 2 Elementary algebra
- 3 Trigonometry
- 4 Areas and volumes
- 5 Polynomials
- 6 Integration and differentiation
- 7 Partial derivatives
- 8 Search this wiki
- 9 See also
- 10 External links
- 11 References
In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. Likewise, if the input is 3, then the output is also 9, and we may write f(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The input variable(s) are sometimes referred to as the argument(s) of the function.
Euclids "common notions"
Things that do not differ from one another are equal to one another
Things that are equal to the same thing are also equal to one another
If equals are added to equals, then the wholes are equal
If equals are subtracted from equals, then the remainders are equal
The whole is greater than the part.
Elementary algebra builds on and extends arithmetic by introducing letters called variables to represent general (non-specified) numbers.
- Added terms are simplified using coefficients. For example, can be simplified as (where 3 is a numerical coefficient).
- Multiplied terms are simplified using exponents. For example, is represented as
- Like terms are added together, for example, is written as , because the terms containing are added together, and, the terms containing are added together.
- Brackets can be "multiplied out", using the distributive property. For example, can be written as which can be written as
- Expressions can be factored. For example, , by dividing both terms by can be written as
For any function , if then:
One must be careful though when squaring both sides of an equation since this can result is solutions that dont satisfy the original equation.
A function is an even function if f(x) = f(-x)
A function is an odd function if f(x) = -f(-x)
The law of sines (also known as the "sine rule") for an arbitrary triangle states:
where is the area of the triangle
The law of tangents:
The parallelogram law reduces to the Pythagorean theorem when the parallelogram is a rectangle
A right triangle is a triangle with gamma=90 degrees.
For small values of x, sin x ≈ x. (If x is in radians).
SOH → sin = "opposite" / "hypotenuse"
CAH → cos = "adjacent" / "hypotenuse"
TOA → tan = "opposite" / "adjacent"
= sin A = a/c
= cos A = b/c
= tan A = a/b
(Note: the expression of tan(x) has i in the numerator, not in the denominator, because the order of the terms (and thus the sign) of the numerator is changed w.r.t. the expression of sin(x).)
- See also: Hyperbolic angle
Hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.
- Hyperbolic sine:
- Hyperbolic cosine:
- Hyperbolic tangent:
- Hyperbolic cotangent:
- Hyperbolic secant:
- Hyperbolic cosecant:
Areas and volumes
The length of the circumference C of a circle is related to the radius r and diameter d by:
- = 3.141592654
- = 2 * π
The area of a circle is:
The surface area of a sphere is
- The surface area of a sphere 1 unit in radius is:
- The surface area of a sphere 128 units in radius is:
The volume inside a sphere is
- The volume of a sphere 1 unit in radius is:
The moment of inertia of a hollow sphere is:
Moment of inertia of a sphere is:
The area of a hexagon is:
- where a is the length of any side.
A square that is twice as big is four times as massive because it is 2 dimensional (22 = 4). A cube that is twice as big is eight times as massive because it is 3 dimensional (23 = 8).
A triangle that is twice as big is four times as massive.
But a Sierpiński triangle that is twice as big is exactly three times as massive. It therefore has a Hausdorff dimension of 1.5849. (21.5849 = 3)
A pyramid that is twice as big is eight times as massive.
But a Sierpiński pyramid that is twice as big is exactly five times as massive. It therefore has a Hausdorff dimension of 2.3219 (22.3219 = 5)
- See also: Runge's phenomenon, Polynomial ring, System of polynomial equations, Rational root theorem, Descartes' rule of signs, and Complex conjugate root theorem
- From Wikipedia:Polynomial:
A polynomial can always be written in the form
- where are constants called coefficients and n is the degree of the polynomial.
- A linear polynomial is a polynomial of degree one.
- Each individual term is the product of the coefficient and a variable raised to a nonnegative integer power.
- A monomial has only one term.
- A binomial has 2 terms.
- Every single-variable, degree n polynomial with complex coefficients has exactly n complex roots.
- However, some or even all of the roots might be the same number.
- A root (or zero) of a function is a value of x for which Z(x)=0.
- If then z2 is a root of multiplicity k. z2 is a root of multiplicity k-1 of the derivative (Derivative is defined below) of Z(x).
- If k=1 then z2 is a simple root.
- The graph is tangent to the x axis at the multiple roots of f and not tangent at the simple roots.
- The graph crosses the x-axis at roots of odd multiplicity and bounces off (not goes through) the x-axis at roots of even multiplicity.
- Near x=z2 the graph has the same general shape as
- The complex conjugate root theorem states that if the coefficients of a polynomial are real, then the non-real roots appear in pairs of the form (a + ib, a – ib).
- The roots of the formula are given by the Quadratic formula:
- is called the discriminant.
- This is a parabola shifted to the right h units, stretched by a factor of a, and moved upward k units.
- k is the value at x=h and is either the maximum or the minimum value.
- The roots of
- are the multiplicative inverses of
- There is no formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc). See Galois theory.
- Where See Binomial coefficient
- Isaac Newton generalized the binomial theorem to allow real exponents other than nonnegative integers. (The same generalization also applies to complex exponents.) In this generalization, the finite sum is replaced by an infinite series.
A monic polynomial is a one variable polynomial in which the leading coefficient is equal to 1.
A rational function is a function of the form
- The vertical asymptotes are the poles of the rational function.
- If n<m then f(x) has a horizontal asymptote at the x axis
- If n=m then f(x) has a horizontal asymptote at k.
- If n>m then f(x) has no horizontal asymptote.
- Given two polynomials and , where the pi are distinct constants and deg Z < m, partial fractions are generally obtained by supposing that
- and solving for the ci constants, by substitution, by equating the coefficients of terms involving the powers of x, or otherwise.
- If the degree of Z is not less than m then use long division to divide P into Z. The remainder then replaces Z in the equation above and one proceeds as before.
- If then
A Generalized hypergeometric series is given by
- where c0=1 and
The function f(x) has n zeros and m poles.
- Basic hypergeometric series, or hypergeometric q-series, are q-analogue generalizations of generalized hypergeometric series.
- We define the q-analog of n, also known as the q-bracket or q-number of n, to be
- Elliptic hypergeometric series are generalizations of basic hypergeometric series.
- An elliptic function is a meromorphic function that is periodic in two directions.
A generalized hypergeometric function is given by
So for ex (see below) we have:
Integration and differentiation
The integral is a generalization of multiplication.
- For example: a unit mass dropped from point x2 to point x1 will release energy.
- The usual equation is is a simple multiplication:
- But that equation cant be used if the strength of gravity is itself a function of x.
- The strength of gravity at x1 would be different than it is at x2.
- And in reality gravity really does depend on x (x is the distance from the center of the earth):
- (See inverse-square law.)
- However, the corresponding Definite integral is easily solved:
The surprisingly simple rules for solving definite integrals
k and y are arbitrary constants:
(Units (feet, mm...) behave exactly like constants.)
And most conveniently :
- The integral of a function is equal to the area under the curve.
- When the "curve" is a constant (in other words, k•x0) then the integral reduces to ordinary multiplication.
The derivative is a generalization of division.
The derivative of the integral of f(x) is just f(x).
The derivative of a function at any point is equal to the slope of the function at that point.
The equation of the line tangent to a function at point a is
The Lipschitz constant of a function is a real number for which the absolute value of the slope of the function at every point is not greater than this real number.
The derivative of f(x) where f(x) = k•xy is
- The derivative of a is
Chain rule for the derivative of a function of a function:
The Chain rule for a function of 2 functions:
- (See "partial derivatives" below)
- (because is negligible)
By the chain rule:
Therefore the Quotient rule:
There is a chain rule for integration but the inner function must have the form so that its derivative and therefore
Actually the inner function can have the form so that its derivative and therefore provided that all factors involving x cancel out.
The product rule for integration is called Integration by parts
The fundamental theorem of Calculus is:
The fundamental theorem of calculus is just the particular case of the Leibniz integral rule:
A differential form is a generalisation of the notion of a differential that is independent of the choice of coordinate system. f(x,y) dx ∧ dy is a 2-form in 2 dimensions (an area element). The derivative operation on an n-form is an n+1-form; this operation is known as the exterior derivative. By the generalized Stokes' theorem, the integral of a function over the boundary of a manifold is equal to the integral of its exterior derivative on the manifold itself.
Taylor & Maclaurin series
- See also: Wikipedia:Volterra series
If we know the value of a smooth function at x=0 (smooth means all its derivatives are continuous) and we also know the value of all of its derivatives at x=0 then we can determine the value at any other point x by using the Maclaurin series. ("!" means factorial)
The proof of this is actually quite simple. Plugging in a value of x=0 causes all terms but the first to become zero. So, assuming that such a function exists, a0 must be the value of the function at x=0. Simply differentiate both sides of the equation and repeat for the next term. And so on.
Because the functions can be multiplied by scalars and added they therefore form an infinite dimensional vector space. (An infinite dimensional space is not a Compact space.) The function f(x) occupies a single point in that infinite dimensional space corresponding to a vector whose components are
The Taylor series generalizes the Maclaurin series.
- The set of germs is a Riemann surface.
- Riemann surfaces are the objects on which multi-valued functions become single-valued.
- The above holds true even if x is a matrix. See Matrix exponential
It then follows that and therefore See Euler's formula
- x is the angle in radians.
- This makes the equation for a circle in the complex plane, and by extension sine and cosine, extremely simple and easy to work with especially with regard to differentiation and integration.
- For sine waves differentiation and integration are replaced with multiplication and division. Calculus is replaced with algebra. Therefore any expression that can be represented as a sum of sine waves can be easily differentiated or integrated.
The Maclaurin series cant be used for a discontinuous function like a square wave because it is not differentiable. (Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. See Generalized function.)
A square wave consists of all odd frequencies with the amplitude of each frequency being .
A dirac comb consists of all integer frequencies with the amplitude of each frequency being 1.
- The reason this works is because sine and cosine are orthogonal functions.
- Two vectors are said to be orthogonal when:
- or more generally when:
- Two functions are said to be orthogonal when:
- That means that multiplying any 2 sine waves of frequency n and frequency m and integrating over one period will always equal zero unless n=m.
- See the graph of sin2(x) to the right.
- And of course ∫ fn*(f1+f2+f3+...) = ∫ (fn*f1) + ∫ (fn*f2) + ∫ (fn*f3) +...
- The complex form of the Fourier series uses complex exponentials instead of sine and cosine and uses both positive and negative frequencies (clockwise and counter clockwise) whose imaginary parts cancel.
- The complex coefficients encode both amplitude and phase and are complex conjugates of each other.
- where the dot between x and ν indicates the inner product of Rn.
- A 2 dimensional Fourier series is used in video compression.
- The FFT has been described as "the most important numerical algorithm of our lifetime"
- In mathematical analysis, many generalizations of Fourier series have proven to be useful.
- They are all special cases of decompositions over an orthonormal basis of an inner product space.
- Spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions (sines and cosines) are used to represent functions on a circle via Fourier series.
- Every continuous function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.
- Every quadratic polynomial can be written as a1+bt+ct2, that is, as a linear combination of the basis functions 1, t, and t2.
Fourier transforms generalize Fourier series to nonperiodic functions like a single pulse of a square wave.
The more localized in the time domain (the shorter the pulse) the more the Fourier transform is spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle.
Using the Fourier transform we can determine that the Dirac delta function consists of all frequencies with the amplitude of each frequency being 1.
- Laplace transforms generalize Fourier transforms to complex frequency .
- Complex frequency includes a term corresponding to the amount of damping.
- , (assuming a > 0)
- The inverse Laplace transform is given by
- The Z-transform can be considered as a discrete-time equivalent of the Laplace transform. This similarity is explored in the theory of time-scale calculus.
- Cauchy kernel =
- Hilbert kernel =
- Poisson Kernel:
- For the ball of radius r, , in Rn, the Poisson kernel takes the form:
- where , (the surface of ), and is the surface area of the unit n-sphere.
- unit disk (r=1) in the complex plane:
- Dirichlet kernel
where denotes point-wise multiplication. It also works the other way around:
By applying the inverse Fourier transform , we can write:
This theorem also holds for the Laplace transform.
where denotes the Fourier transform.
Since sgn(x) = sgn(2πx), it follows that this result applies to the three common definitions of .
By Euler's formula,
Therefore, H(u)(t) has the effect of shifting the phase of the negative frequency components of u(t) by +90° (π/2 radians) and the phase of the positive frequency components by −90°.
And i·H(u)(t) has the effect of restoring the positive frequency components while shifting the negative frequency ones an additional +90°, resulting in their negation.
In electrical engineering, the convolution of one function (the input signal) with a second function (the impulse response) gives the output of a linear time-invariant system (LTI).
At any given moment, the output is an accumulated effect of all the prior values of the input function
- See also: Variation of parameters
where m is the inertial mass, x is its displacement from the equilibrium, and k is the spring constant.
Solving for x produces
A is the amplitude (maximum displacement from the equilibrium position), is the angular frequency, and φ is the phase.
Energy passes back and forth between the potential energy in the spring and the kinetic energy of the mass.
The important thing to note here is that the frequency of the oscillation depends only on the mass and the stiffness of the spring and is totally independent of the amplitude.
That is the defining characteristic of resonance.
V is the voltage, R is the resistance, L is the inductance, C is the capacitance.
I = dQ/dt is the current.
It makes no difference whether the current is a small number of charges moving very fast or a large number of charges moving slowly.
In reality the latter is the case.
If V(t)=0 then the only solution to the equation is the transient response which is a rapidly decaying sine wave with the same frequency as the resonant frequency of the circuit.
- Like a mass (inductance) on a spring (capacitance) the circuit will resonate at one frequency.
- Energy passes back and forth between the capacitor and the inductor with some loss as it passes through the resistor.
If V(t)=sin(t) from -∞ to +∞ then the only solution is a sine wave with the same frequency as V(t) but with a different amplitude and phase.
If V(t) is zero until t=0 and then equals sin(t) then I(t) will be zero until t=0 after which it will consist of the steady state response plus a transient response.
Starting with a linear homogeneous differential equation with constant coefficients ,
it can be seen that if , each term would be a constant multiple of . This results from the fact that the derivative of the exponential function is a multiple of itself. Therefore, , , and are all multiples. This suggests that certain values of will allow multiples of to sum to zero, thus solving the homogeneous differential equation. In order to solve for , one can substitute and its derivatives into the differential equation to get
Since can never equate to zero, it can be divided out, giving the characteristic equation
By solving for the roots, , in this characteristic equation, one can find the general solution to the differential equation. For example, if is found to equal to 3, then the general solution will be , where is an arbitrary constant.
- See also: Currying
The partial derivative with respect to one variable is found by simply treating all other variables as though they were constants.
The "partial" integral can be taken with respect to x (treating y as constant, in a similar manner to partial differentiation):
- Here, the "constant" of integration is no longer a constant, but instead a function of all the variables of the original function except x. The reason for this is that all the other variables are treated as constant when taking the partial derivative, so any function which does not involve will disappear when taking the partial derivative, and we have to account for this when we take the antiderivative.
Let f(x, y, z) be a scalar function (for example electric potential energy or temperature).
- A 2 dimensional example of a scalar function would be an elevation map.
- (Contour lines of an elevation map are an example of a level set.)
And the differential is
Gradient of scalar field
The Gradient of f(x, y, z) is a vector field whose value at each point is a vector (technically its a covector because it has units of distance−1) that points "downhill" with a magnitude equal to the slope of the function at that point.
You can think of the gradient as how much the function changes per unit distance. That sounds like the gradient would therefore be a covector and indeed some people say it is a covector. But those people are wrong. It is a vector. Maybe it should be a covector but it is a vector. Its dual is a covector.
The gradient of temperature gives heat flow.
For static (unchanging) fields the Gradient of the electric potential is the electric field itself. Image to the right shows the potential of a single point charge.
Its gradient gives the electric field which is shown in the 2 images below. In the image on the left the field strength is proportional to the length of the vectors. In the image on the right the field strength is proportional to the density of the flux lines. The image is 2 dimensional and therefore the flux density in the image follows an inverse first power law but in reality the field lines from a real proton or electron spread outward in 3 dimensions and therefore follow an inverse square law. Inverse square means that at twice the distance the field is four times weaker.
The field of 2 point charges is simply the linear sum of the separate charges.
Strictly speaking what we are doing here is finding the derivative with respect to the Cartesian coordinate system. In flat Euclidean space the result is a vector that does not depend on which particular Cartesian coordinate system we use so we dont even bother to specify which one we used.
On a manifold things are not so simple. On a manifold we dont find the derivative with respect to a coordinate system. We find the derivative with respect to a vector (i.e. directional derivative) or a vector field (Lie derivative) on that manifold.
The Divergence of a vector field is a scalar.
The divergence of the electric field is non-zero wherever there is electric charge and zero everywhere else.
Field lines begin and end at charges because the charges create the electric field.
The Laplacian is the divergence of the gradient of a function:
- elliptic operators generalize the Laplacian.
- See also: Biot–Savart law
The Curl of a vector field describes how much the vector field is twisted.
(The field may even go in circles.)
In 3 dimensions the dual of the current vector is a bivector.
In 2 dimensions this reduces to a single scalar
The curl of the gradient of any scalar field is always zero.
See also: differential forms.
Dual of the curl
Green's theorem states that if you want to know how many field lines exit a region then you can either count how many lines cross the boundary (perform a line integral) or you can simply count the number of charges (or the amount of current) within that region. See Divergence theorem.
In 2 dimensions this is
You can think of each field line as ending in a single unit of charge.
A version of Green's theorem also works for curl.
Green's theorem is perfectly obvious when dealing with vector fields but is much less obvious when applied to complex valued functions in the complex plane.
See also Kelvin–Stokes theorem
The complex plane
- Highly recomend: Fundamentals of complex analysis with applications to engineering and science by Saff and Snider
- External link: http://www.solitaryroad.com/c606.html
The formula for the derivative of a complex function f at a point z0 is the same as for a real function:
Every complex function can be written in the form
Because the complex plane is two dimensional, z can approach z0 from an infinite number of different directions.
However, if within a certain region, the function f is holomorphic (that is, complex differentiable) then, within that region, it will only have a single derivative whose value does not depend on the direction in which z approaches z0 despite the fact that fx and fy each have 2 partial derivatives. One in the x and one in the y direction..
This is only possible if the Cauchy–Riemann conditions are true.
An entire function, also called an integral function, is a complex-valued function that is holomorphic at all finite points over the whole complex plane.
As with real valued functions, a line integral of a holomorphic function depends only on the starting point and the end point and is totally independant of the path taken.
The starting point and the end point for any loop are the same. This, of course, implies Cauchy's integral theorem for any holomorphic function f:
Therefore curl and divergence must both be zero for a function to be holomorphic.
Green's theorem for functions (not necessarily holomorphic) in the complex plane:
Next section: Intermediate mathematics/Discrete mathematics
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dx = dy/y = 1/y * dy
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