**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. *f*is 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=(y _{2}-y_{1})/(x_{2}-x_{1}), x_{1} != x_{2}*

*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+zkx=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)=<u_{i} + v_{i}, u_{j} + v_{j}> |

Scalar Product | cu(t)=<cu_{i},cu_{j}> |

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-y_{1}=m(x-x_{1}) |

Slope-intercept Form | y=mx+b |

Where A, B, C, y_{1}, x_{1}a and b are coordinates; m is the slope; x and y are variables.

**Trigonometric Functions**

Euler's Formula | e^{ix}=cos(x)+isin(x) |

Pythagorean Identity | 1=sin^{2}(x)+cos^{2}(x) |

Where x is a variable.

**Log Functions**

General Form | ln x = ∫^{b}_{a} (1/t) dt, for x>0 |

1 | ln(1) = 0 |

Multiple | ln(ab) = ln(a) + ln(b) |

Power | ln(a^{n}) = 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 | e^{x} = f^{-1}[ln(x)] |

Inverse 1 | ln(e^{x}) = x |

Inverse 2 | e^{ln(x)} = x |

Multiple | e^{a}e^{b}=e^{a+b} |

Division | e^{a}/e^{b}=e^{a-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!)f^{n}(c)(x-c)^{n} = f(c) + f'(c)(x-c) + ... + (1/n!)f^{n}(c)(x-c)^{n} + ...*

A Taylor series converges iff the error *R* converges to zero.

lim_{n→∞} R_{n}(x) = lim_{n→∞} [1/(n+1)!]f^{n+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) = a _{0}/2 + Σ^{∞}_{n=1} [a_{n}cos(nπx/T) + b_{n}sin(nπx/T)]*,

where

*a _{n} = (1/T)∫^{T}_{-T}f(x) cos(nπx/T) dx* and

*b _{n} = (1/T)∫^{T}_{-T}f(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) ) det_{Jm} 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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