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Calculus: Functions (Part 1 of 5)
Preface
This series is aimed at providing tools for an electrical engineer to analyze data and solve problems in design. The focus is on applying calculus to equations or physical systems.
Introduction
This article will introduce functions.
There are many calculus references, the one I like to use is Calculus by Larson, Edwards and Hostetler.
This also assumes you are familiar with Python or can stumble your way through it.
Concepts
Number: A mathematical object or symbol used to count, measure and optionally label (interpretation).
Vector: A type of number that indicates direction and magnitude. A vector has an origin and destination and therefore direction and magnitude.
Variable: a symbol used to hold a value or coordinate.
Set: a collection of numbers or variables (see below). In the form of (1,2,3,...,n) or (a,b,c,...,n) and often (x,y) and (x,y,z).
Number System: A set of numbers, often subjected to specific rules.
Coordinate: a particular value taken on by a set used to indicate the position of a point, line, or plane.
Coordinate System: usually a method of representing coordinates geometrically. However a coordinate can be represented in any coordinate system and not necessarily one that has a physical representation, in other words a coordinate system is a set of variables.
Cartesian Coordinate System: a rectilinear system in which the horizontal line on a graph is the x axis and the vertical is the y axis of a plane. This can be extended to three dimensions by including the z axis to represent space extending out from the x and y axis, perpendicular to both.
Polar/Spherical Coordinate System: a coordinate system that is represented by a) distance from origin, b) angle from 0 degrees (counterclockwise) and in three dimensions c) the angle from 0 degrees (and also perpendicular to both a and b).
Function: a real-valued function f maps X onto Y such that for every coordinate in X there is one and one only output on Y; where X and Y are sets of real numbers such that X=(a1,b1,c1,...,z1) and Y=(a2,b2,c2,...,z2).
Vector Valued Function: describes the magnitude and direction of a vector originating at the origin (0,0) and terminating at the evaluation of r(t).
Vector Field: describes a plane or space on which a vector maps to every point in the space. In other words each point in the plane or space has a vector of specific magnitude and direction.
Conservative Vector Field: describes a vector field F such that F=∇ f. In other words F is a gradient. fis referred to as the potential function.
Solution: a function ψ that when substituted in function θ(ψ) satisfies the equation for all ψ in the interval I.
Intercept: a coordinate at which one coordinate value in the set is zero (e.g. f(a,b,0,d) or g(0,b,c)).
Intersection: a coordinate at which two sets of data (or functions) intercept.
Graph: a visual representation of coordinate sets, most often in 2 to 4 variables.
Slope: rate of change of a function. For example the slope of a straight line: m=(y2-y1)/(x2-x1), x1 != x2
Homogeneous Function: a function of the form F(x) = 0.
Nonhomogeneous Function: a function of the form F(x) = G.
Inverse: A function g is the inverse of the function f if f(g(x)) = x for each x in the domain of g AND g(f(x)) = x for each x in the domain of f. All inverse functions are reflective about the line y=x.
Sequence: A set of numbers following a pattern or following a specific set of rules (e.g. sequence in time, boundary conditions, etc). They may be infinite in length.
Series: The summation of an infinitely long sequence.
Vector and Parametric Functions and Properties
| General Form: Vector Valued Function | r(t)=f(t)i+g(t)j+h(t)k |
| General Form: Vector Field | F(x,y)=xi+y j+zk x=M(x,y,z), y=N(x,y,z), z=P(x,y,z) |
| General Form: Parametric Form | x=M(t), y=N(t), z=P(t) |
| Addition | u(t) + v(t)=<ui + vi, uj + vj> |
| Scalar Product | cu(t)=<cui,cuj> |
| Dot Product | |
| Cross Product | |
| Commutative | u + v=v + u |
| Associative | (u+v)+w=u+(v+w) |
| Additive Identity | u+0=u |
| Additive Inverse | u+(-u)=0 |
| Distributive | (c+d)u=cu+du |
| Distributive 2 | c(u+v)=cu+cv |
| Scalar | 1(u)=u |
| Scalar 2 | 0(u)=0 |
| Curl | |
| Divergence | |
| Step Function | cu(t-a)={0, t-a<0; c, 0>t-a} |
| Periodic Function | f(t+T) = f(t) |
| Dirac Delta Function | δ={0, t != 0; ∞, t=0} |
Where f, g and h are functions of t; r is a vector function of t; and i, j and k are vectors.
All the properties of the functions below can be components of a vector function. Therefore all the properties of the above properties apply to the functions below.
Linear Functions
| General Form | Ax+By+C=0 |
| Vertical Line | x=a |
| Horizontal Line | y=b |
| Point-slope Form | y-y1=m(x-x1) |
| Slope-intercept Form | y=mx+b |
Where A, B, C, y1, x1a and b are coordinates; m is the slope; x and y are variables.
Trigonometric Functions
| Euler's Formula | eix=cos(x)+isin(x) |
| Pythagorean Identity | 1=sin2(x)+cos2(x) |
Where x is a variable.
Log Functions
| General Form | ln x = ∫ba (1/t) dt, for x>0 |
| 1 | ln(1) = 0 |
| Multiple | ln(ab) = ln(a) + ln(b) |
| Power | ln(an) = n ln a |
| Division | ln(a/b) = ln(a) - ln(b) |
Where a, b and n are coordinates; x is a variable.
Exponential Functions
| General Form | ex = f-1[ln(x)] |
| Inverse 1 | ln(ex) = x |
| Inverse 2 | eln(x) = x |
| Multiple | eaeb=ea+b |
| Division | ea/eb=ea-b |
Where a, b and n are coordinates; x is a variable.
Function Transformations
| Original Graph (Reference) | y=f(x) |
| Horizontal Shift | y=f(x±c) |
| Vertical Shift | y=f(x)±c |
| Reflection (about x) | y=-f(x) |
| Reflection (about y) | y=f(-x) |
| Reflection (about origin) | y=-f(-x) |
Function Expansion: Taylor Series
Decompose a function into a series of power terms.
If a function f has derivatives of all orders at x=c, then the Taylor series of f is
f(x) = ∑n=0∞(1/n!)fn(c)(x-c)n = f(c) + f'(c)(x-c) + ... + (1/n!)fn(c)(x-c)n + ...
A Taylor series converges iff the error R converges to zero.
limn→∞ Rn(x) = limn→∞ [1/(n+1)!]fn+1(z)(x-c)n+1 = 0
Function Expansion: Fourier Series
Decompose a function into a series of sines and/or cosines.
If f is a piecewise continuous function on the internal [-T,T], then the Fourier series of f is
f(x) = a0/2 + Σ∞n=1 [ancos(nπx/T) + bnsin(nπx/T)],
where
an = (1/T)∫T-Tf(x) cos(nπx/T) dx and
bn = (1/T)∫T-Tf(x) sin(nπx/T) dx.
A Fourier series converges if f'(x) is continuous on the interval.
Coordinate System Transformations
For any two coordinate systems in which the conversion between the two is defined, you can use a Jacobian Matrix to determine the conversion factor to define a change of variables.
If x=g(u,v) and y=h(u,v), then the Jacobian of x and y with respect to u and v is:
The Jacobian is defined here (not easy to represent in HTML)
The change of variables for a double integral is then defined as:
∫R∫ f(x,y) dx dy = ∫S ∫ f( g(u,v), h(u,v) ) detJm du dv
The definition above is for two variables, but can be extended to more variables with a larger Jacobian/number of integrals and also reduced to a single variable/integral.
Axis Intercepts
To determine where a function crosses the x or y axis evaluate the following function:
f(0) (for y-intercepts)
f(x)=0 (for x-intercepts)
If the above formula fails try using Newton's Method (not illustrated here).
Next Up
Continuity and Limits of functions.
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