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Calculus/Functions

Functions are everywhere, from a simple correlation between distance and time to complex heat waves. This chapter focuses on the fundamentals of functions: the definition, basic concepts, and other defining aspects. It is very concept-heavy, and expect a lot of reading and understanding. However, this is simply a review and an introduction on what is to come in future chapters.

Introduction
Whenever one quantity uniquely determines the value of another quantity, we have a function. That is, the set $$X$$ uniquely determines the set $$Y$$. You can think of a function as a kind of machine. You feed the machine raw materials, and the machine changes the raw materials into a finished product.

Think about dropping a ball from a bridge. At each moment in time, the ball is a height above the ground. The height of the ball is a function of time. It was the job of physicists to come up with a formula for this function. This type of function is called real-valued since the "finished product" is a number (or, more specifically, a real number).

Think about a wind storm. At different places, the wind can be blowing in different directions with different intensities. The direction and intensity of the wind can be thought of as a function of position. This is a function of two real variables (a location is described by two values - an $$x$$ and a $$y$$) which results in a vector (which is something that can be used to hold a direction and an intensity). These functions are studied in multivariable calculus (which is usually studied after a one year college level calculus course). This a vector-valued function of two real variables.

We will be looking at real-valued functions until studying multivariable calculus. Think of a real-valued function as an input-output machine; you give the function an input, and it gives you an output which is a number (more specifically, a real number). For example, the squaring function takes the input 4 and gives the output value 16. The same squaring function takes the input -1 and gives the output value 1.

Notation
Functions are used so much that there is a special notation for them. The notation is somewhat ambiguous, so familiarity with it is important in order to understand the intention of an equation or formula.

Though there are no strict rules for naming a function, it is standard practice to use the letters $$f$$, $$g$$ , and $$h$$ to denote functions, and the variable $$x$$ to denote an independent variable. $$y$$ is used for both dependent and independent variables.

When discussing or working with a function $$f$$, it's important to know not only the function, but also its independent variable $$x$$. Thus, when referring to a function $$f$$, you usually do not write $$f$$, but instead $$f(x)$$. The function is now referred to as "$$f$$ of $$x$$". The name of the function is adjacent to the independent variable (in parentheses). This is useful for indicating the value of the function at a particular value of the independent variable. For instance, if
 * $$f(x)=7x+1$$ ,

and if we want to use the value of $$f$$ for $$x$$ equal to $$2$$, then we would substitute 2 for $$x$$ on both sides of the definition above and write
 * $$f(2)=7(2)+1=14+1=15$$

This notation is more informative than leaving off the independent variable and writing simply '$$f$$', but can be ambiguous since the parentheses next to $$f$$ can be misinterpreted as multiplication, $$2f$$. To make sure nobody is too confused, follow this procedure:

 * 1) Define the function $$f$$ by equating it to some expression.
 * 2) Give a sentence like the following: "At $$x=c$$, the function $$f$$ is..."
 * 3) Evaluate the function.

Description
There are many ways which people describe functions. In the examples above, a verbal description is given (the height of the ball above the earth as a function of time). Here is a list of ways to describe functions. The top three listed approaches to describing functions are the most popular.

 * 1) A function is given a name (such as $$f$$) and a formula for the function is also given. For example, $$f(x)=3x+2$$ describes a function. We refer to the input as the argument of the function (or the independent variable), and to the output as the value of the function at the given argument.
 * 2) A function is described using an equation and two variables. One variable is for the input of the function and one is for the output of the function. The variable for the input is called the independent variable. The variable for the output is called the dependent variable. For example, $$y=3x+2$$ describes a function. The dependent variable appears by itself on the left hand side of equal sign.
 * 3) A verbal description of the function.

When a function is given a name (like in number 1 above), the name of the function is usually a single letter of the alphabet (such as $$f$$ or $$g$$). Some functions whose names are multiple letters (like the sine function $$y=\sin(x)$$)

If we write $$f(x)=3x+2$$, then we know that

 * The function $$f$$ is a function of $$x$$.
 * To evaluate the function at a certain number, replace the $$x$$ with that number.
 * Replacing $$x$$ with that number in the right side of the function will produce the function's output for that certain input.
 * In English, the definition of $$f$$ is interpreted, "Given a number, $$f$$ will return two more than the triple of that number."

How would we know the value of the function $$f$$ at 3? We would have the following three thoughts:

and we would write
 * 1) $$f(3)=3(3)+2$$
 * 2) $$3(3)+2=9+2$$
 * 3) $$9+2=11$$

$$f(3)=3(3)+2=9+2=11$$.

The value of $$f$$ at 3 is 11.

Note that $$f(3)$$ means the value of the dependent variable when $$x$$ takes on the value of 3. So we see that the number 11 is the output of the function when we give the number 3 as the input. People often summarize the work above by writing "the value of $$f$$ at three is eleven", or simply "$$f$$ of three equals eleven".

Basic concepts of functions
The formal definition of a function states that a function is actually a mapping that associates the elements of one set called the domain of the function, $$A$$, with the elements of another set called the range of the function, $$B$$. For each value we select from the domain of the function, there exists exactly one corresponding element in the range of the function. The definition of the function tells us which element in the range corresponds to the element we picked from the domain. An example of how is given below.

In mathematics, it is important to notice any repetition. If something seems to repeat, keep a note of that in the back of your mind somewhere.

Here, notice that $$f(x)=5x+\frac{5x}{2}$$ and $$f(x)=225\frac{1}{2}$$. Because $$f(x)$$ is equal to two different things, it must be the case that the other side of the equal sign to $$f(x)$$ is equal to the other. This property is known as the transitive property and could thus make the following equation below true:
 * $$225\frac{1}{2}=5x+\frac{5x}{2}$$

Next, simplify — make your life easier rather than harder. In this instance, since $$5x+\frac{5x}{2}$$ has $$x$$ as a like-term, and the two terms are fractions added to the other, put it over a common denominator and simplify. Similar, since $$225\frac{1}{2}$$ is a mixed fraction, $$225\frac{1}{2}=225+\frac{1}{2}$$.
 * $$225\frac{1}{2}=5x+\frac{5x}{2}$$
 * $$\Rightarrow\frac{225}{1}+\frac{1}{2}=\frac{5x}{1}+\frac{5x}{2}$$
 * $$\Rightarrow\frac{450}{2}+\frac{1}{2}=\frac{10x}{2}+\frac{5x}{2}$$
 * $$\Rightarrow\frac{451}{2}=\frac{15x}{2}$$

Multiply both sides by the reciprocal of the coefficient of $$x$$, $$\frac{2}{15}$$ from both sides by multiplying by it.
 * $$\frac{2}{15}\cdot\frac{451}{2}=x$$
 * $$\frac{451}{15}=x$$ or $$x=\frac{451}{15}$$.

The value of $$x$$ that makes $$f(x)=225\frac{1}{2}$$ is $$x=\frac{451}{15}$$.$$\blacksquare$$.

Classically, the element picked from the domain is pictured as something that is fed into the function and the corresponding element in the range is pictured as the output. Since we "pick" the element in the domain whose corresponding element in the range we want to find, we have control over what element we pick and hence this element is also known as the "independent variable". The element mapped in the range is beyond our control and is "mapped to" by the function. This element is hence also known as the "dependent variable", for it depends on which independent variable we pick. Since the elementary idea of functions is better understood from the classical viewpoint, we shall use it hereafter. However, it is still important to remember the correct definition of functions at all times.

Basic types of transformation
To make it simple, for the function $$f(x)$$, all of the possible $$x$$ values constitute the domain, and all of the values $$f(x)$$ ($$y$$ on the x-y plane) constitute the range. To put it in more formal terms, a function $$f$$ is a mapping of some element $$a\in A$$, called the domain, to exactly one element $$b\in B$$, called the range, such that $$f:A\to B$$. The image below should help explain the modern definition of a function:
 * 1) A function is considered one-to-one if an element $$a\in A$$ from domain $$A$$ of function $$f$$, leads to exactly one element $$b\in B$$ from range $$B$$ of the function. By definition, since only one element $$b$$ is mapped by function $$f$$ from some element $$a$$, $$f:A\to B$$ implies that there exists only one element $$b$$ from the mapping. Therefore, there exists a one-to-one function because it complies with the definition of a function. This definition is similar to Figure 1.
 * 2) A function is considered many-to-one if some elements $$a_{1},a_{2},\ldots a_{n}\in A$$ from domain $$A$$ of function $$f$$ maps to exactly one element $$b\in B$$ from range $$B$$ of the function. Since some elements $$\left( a_{1},a_{2},\ldots a_{n} \right)\in A$$ must map onto exactly one element $$b\in B$$, $$f:A\to B$$ must be compliant with the definition of a function. Therefore, there exists a many-to-one function.
 * 3) A function is considered one-to-many if exactly one element $$a\in A$$ from domain $$A$$ of function $$f$$ maps to some elements $$\left( b_{1},b_{2},\ldots b_{n} \right)\in B$$ from range $$B$$ of the function. If some element $$a$$ maps onto many distinct elements $$b$$, then $$f:A\to B$$ is non-functional since there exists many distinct elements $$b$$. Given many-to-one is non-compliant to the definition of a function, there exists no function that is one-to-many.

The modern definition describes a function sufficiently such that it helps us determine whether some new type of "function" is indeed one. Now that each case is defined above, you can now prove whether functions are one of these function cases. Here is an example problem:

Notice that the only changing element in the function $$f$$ is $$x$$. To prove a function is one-to-one by applying the definition may be impossible because although two random elements of domain set $$A$$ may not be many-to-one, there may be some elements $$\left( a_{1},a_{2},\ldots a_{n} \right)\in A$$ that may make the function many-to-one. To account for this possibility, we must prove that it is impossible to have some result like that.

Assume there exists two distinct elements $$x_{1}\ne x_2$$ that will result in $$f\left( x_{1} \right) =f\left( x_{2} \right)$$. This would make the function many-to-one. In consequence, $$ax_{1}+b=ax_{2}+b$$

Subtract $$b$$ from both sides of the equation.
 * $$\Rightarrow ax_{1}=ax_{2}$$

Subtract $$ax_{2}$$ from both sides of the equation.
 * $$\Rightarrow ax_{1}-ax_{2}=0$$

Factor $$a$$ from both terms to the left of the equation.
 * $$\Rightarrow a\left( x_{1}-x_{2} \right) =0$$

Multiply $$\frac{1}{a}$$ to both sides of the equation.
 * $$\Rightarrow \left( x_{1}-x_{2} \right) =0$$

Add $$x_{2}$$ to both sides of the equation.
 * $$\Rightarrow x_{1}=x_{2}$$

Notice that $$x_{1}=x_{2}$$. However, this is impossible because $$x_{1}$$ and $$x_{2}$$ are distinct. Ergo, $$f\left( x_{1} \right)\ne f\left( x_{2} \right)$$. No two distinct inputs can give the same output. Therefore, the function must be one-to-one.

Basic concepts
[[File:Domain, Range, Codomain.svg|thumb|230x230px|The domain is all the elements in set $$A$$ that can be mapped to the elements in set $$B$$.

The range is those elements in set $$B$$ which are mapped with the domain.

The codomain is all the elements in set $$B$$.]]There are a few more important ideas to learn from the modern definition of the function, and it comes from knowing the difference between domain, range, and codomain. We already discussed some of these, yet knowing about sets adds a new definition for each of the following ideas. Therefore, let us discuss these based on these new ideas. Let $$A$$ and $$B$$ be a set. If we were to combine these two sets, it would be defined as the cartesian cross product $$A\times B$$. The subset of this product is the function. The below definitions are a little confusing. The best way to interpret these is to draw an image. To the right of these definitions is the image that corresponds to it.

Note that the codomain is not as important as the other two concepts.

Take $$f(x)=\sqrt{1-x^2}$$ for example:Because of the square root, the content in the square root should be strictly not smaller than 0."$1-x^2\geq0$""$\Leftrightarrow -1\leq x\leq 1$"Thus the domain is"$x\in[-1,1]$"

Correspondingly, the range will be"$f(x)\in[0,1]$"

Other types of transformation
There is one more final aspect that needs to be defined. We already have a good idea of what makes a mapping a function (e.g. it cannot be one-to-many). However, three more definitions that you will often hear will be necessary to talk about: injective, surjective, bijective.

 * A function is injective if it is one-to-one.
 * A function is surjective if it is "onto." That is, the mapping $$f:A\to B$$ has $$B$$ as the range of the function, where the codomain and the range of the function are the same.
 * A function is bijective if it is both surjective and injective.

Again, the above definitions are often very confusing. Again, another image is provided to the right of the bullet points. Along with that, another example is also provided. Let us analyze the function $$g(x)=ax^{2}$$. Notice that the only changing element in the function $$g$$ is $$x$$. Let us check to see the conditions of this function.

Is it injective? Let $$g(x)=y$$, and solve for $$x$$. First, divide by $$a$$.
 * $$y=ax^{2}$$
 * $$\Rightarrow \frac{y}{a}=x^{2}$$

Then take the square root of $$x^2$$. By definition, $$\sqrt{x^{2}}=\pm\sqrt{x^{2}}=\pm x$$, so
 * $$\frac{y}{a}=x^{2}$$
 * $$\Rightarrow\pm\sqrt{\frac{y}{a}}=\pm x$$

Notice, however, what we learned from the above manipulation. As a result of solving for $$x$$, we found that there are two solutions for $$x$$. However, this resulted in two different values from $$\frac{y}{a}$$. This means that for some individual $$x$$ that gives $$y$$, there are two different inputs that result in the same value. Because $$f(x_{1})=f(x_{2})$$ when $$x_{1}\ne x_{2}$$, this is by definition non-injective.

Is it surjective? As a natural consequence of what we learned about inputs, let us determine the range of the function. After all, the only way to determine if something is surjective is to see if the range applies to all real numbers. A good way to determine this is by finding a pattern involving our domains. Let us say we input a negative number for the domain: $$f(-x)=a(-x)^{2}=a\cdot(-x)\cdot(-x)=ax^2$$. This seemingly trivial exercise tells us that negative numbers give us non-negative numbers for our range. This is huge information! For $$x\in\mathbb{R}$$, the function always results $$y>0$$ for our range. For the set $$B$$, we have elements in that set that have no mappings from the set $$A$$. As such, $$y\le0$$ is the codomain of set $$B$$. Therefore, this function is non-surjective!

The vertical line test
The vertical line test is a systematic test to find out if an equation involving $$x$$ and $$y$$ can serve as a function (with $$x$$ the independent variable and $$y$$ the dependent variable). Simply graph the equation and draw a vertical line through each point of the $$x$$-axis. If any vertical line ever touches the graph at more than one point, then the equation is not a function; if the line always touches at most one point of the graph, then the equation is a function.

The circle, on the right, is not a function because the vertical line intercepts two points on the graph when $$x=1$$.

The horizontal line and the algebraic 1-1 test
Similarly, the horizontal line test, though does not test if an equation is a function, tests if a function is injective (one-to-one). If any horizontal line ever touches the graph at more than one point, then the function is not one-to-one; if the line always touches at most one point on the graph, then the function is one-to-one.

The algebraic 1-1 test is the non-geometric way to see if a function is one-to-one. The basic concept is that:

Assume there is a function $$f$$. If:"$f(a)=f(b)$, and $a=b$, then"function $$f$$ is one-to-one.

Here is an example: prove that $$f(x)=\frac{1-2x}{1+x}$$ is injective.

Since the notation is the notation for a function, the equation is a function. So we only need to prove that it is injective. Let $$a$$ and $$b$$ be the inputs of the function and that $$f(a)=f(b)$$. Thus,

 * $$\frac{1-2a}{1+a}=\frac{1-2b}{1+b}$$
 * $$\Leftrightarrow(1+b)(1-2a)=(1+a)(1-2b)$$
 * $$\Leftrightarrow1-2a+b-2ab=1-2b+a-2ab$$ $$\Leftrightarrow1-2a+b=1-2b+a$$
 * $$\Leftrightarrow1-2a+3b=1+a$$
 * $$\Leftrightarrow1+3b=1+3a$$
 * $$\Leftrightarrow a=b$$

So, the result is $$a=b$$, proving that the function is injective.

Another example is proving that $$g(x)=x^2$$ is not injective.

Using the same method, one can find that $$a=\pm b$$, which is not $$a=b$$. So, the function is not injective.

Remarks
The following arise as a direct consequence of the definition of functions: Functions are an important foundation of mathematics. This modern interpretation is a result of the hard work of the mathematicians of history. It was not until recently that the definition of the relation was introduced by René Descartes in Geometry (1637). Nearly a century later, the standard notation ($$f(x)=y$$) was first introduced by Leonhard Euler in Introductio in Analysin Infinitorum and Institutiones Calculi Differentialis. The term function was also a new innovation during Euler's time as well, being introduced Gottfried Wilhelm Leibniz in a 1673 letter "to describe a quantity related to points of a curve, such as a coordinate or curve's slope." Finally, the modern definition of the function being the relation among sets was first introduced in 1908 by Godfrey Harold Hardy where there is a relation between two variables $$x$$ and $$y$$ such that "to some values of $$x$$ at any rate correspond values of $$y$$." For the person that wants to learn more about the history of the function, go to History of functions.
 * 1) By definition, for each "input" a function returns only one "output", corresponding to that input. While the same output may correspond to more than one input, one input cannot correspond to more than one output. This is expressed graphically as the vertical line test: a line drawn parallel to the axis of the dependent variable (normally vertical) will intersect the graph of a function only once. However, a line drawn parallel to the axis of the independent variable (normally horizontal) may intersect the graph of a function as many times as it likes. Equivalently, this has an algebraic (or formula-based) interpretation. We can always say if $$a=b$$, then $$f(a)=f(b)$$ , but if we only know that $$f(a)=f(b)$$ then we can't be sure that $$a=b$$.
 * 2) Each function has a set of values, the function's domain, which it can accept as input. Perhaps this set is all positive real numbers; perhaps it is the set {pork, mutton, beef}. This set must be implicitly/explicitly defined in the definition of the function. You cannot feed the function an element that isn't in the domain, as the function is not defined for that input element.
 * 3) Each function has a set of values, the function's range, which it can output. This may be the set of real numbers. It may be the set of positive integers or even the set {0,1}. This set, too, must be implicitly/explicitly defined in the definition of the function.

Important functions
The functions listed below are essential to calculus. This section only serves as a review and scratches the surface of those functions. If there are any questions about those functions, please review the materials related to those functions before continuing. More about graphing will be explained in Chapter

Polynomials
Polynomial functions are the most common and most convenient functions in the world of calculus. Their frequent appearances and their applications on computer calculations have made them important.

Constant
When $$n=0$$, the polynomial can be rewritten into the following function:"$f(x)=C$, where $C$ is a constant."The graph of this function is a horizontal line passing the point $$(0,C)$$.

Linear
When $$n=1$$, the polynomial can be rewritten into"$f(x)=mx+b$, where $m\text{ and }b$ are constants."The graph of this function is a straight line passing the point $$(0,b)$$ and $$(-\frac{b}{m},0)$$, and the slope of this function is $$m$$.

Quadratic
When $$n=2$$, the polynomial can be rewritten into"$f(x)=ax^2+bx+c$, where $a,b,c$ are constants."The graph of this function is a parabola, like the trajectory of a basketball thrown into the hoop.

If there are questions about the quadratic formula and other basic polynomial concepts, please review the respective chapters in Algebra.

Trigonometric
Trigonometric functions are also very important because it can connect algebra and geometry. Trigonometric functions are explained in detail here due to its importance and difficulty.

Exponential and Logarithmic
Exponential and logarithmic functions have a unique identity when calculating the derivatives, so this is a great time to review those functions.

A special number will be frequently seen in those functions: the Euler's constant, also known as the base of the natural logarithm. Notated as $$e$$, it is defined as $$e=\sum_{n=0}^\infty \frac{1}{n!}$$.

Signum
The Signum (sign) function is simply defined as{{blockquote|text=$$\mbox{sgn}(x)=\left\{\begin{matrix}-1 & \text{if} & x<0\\0 & \text{if} & x=0 \\ 1 & \text{if} & x>0 \end{matrix} \right.$$}}

Properties of functions
Sometimes, a lot of function manipulations cannot be achieved without some help from basic properties of functions.

Domain and range
This concept is discussed above.

Growth
Although it seems obvious to spot a function increasing or decreasing, without the help of graphing software, we need a mathematical way to spot whether the function is increasing or decreasing, or else our human minds cannot immediately comprehend the huge amount of information.

Assume a function $$f(x)$$ with inputs $$x_1,x_2$$, and that $$x_1\in[a,b]$$, $$x_2\in[a,b]$$, and $$x_2>x_1$$ at all times. If for all $$x_1$$ and $$x_2$$, $$f(x_2)-f(x_1)>0$$, then

$$f(x)$$ is increasing in $$[a,b]$$

If for all $$x_1$$ and $$x_2$$, $$f(x_2)-f(x_1)<0$$, then

$$f(x)$$ is decreasing in $$[a,b]$$ Example: In which intervals is $$f(x)=\frac{1}{1-x^2}$$ increasing?

Firstly, the domain is important. Because the denominator cannot be 0, the domain for this function is

$$x\in(-\infty,-1)\cup(-1,1)\cup(1,\infty)$$

In $$(-\infty,-1)$$, the growth of the function is: Let $$x_1,x_2\in[-\infty,-1]$$ and $$x_2>x_1$$

Thus,"$f(x_2)-f(x_1)=\frac{1}{1-x_2^2}-\frac{1}{1-x_1^2}=\frac{x_2^2-x_1^2}{(1-x_2^2)(1-x_1^2)}$"$$\because$$ both $$x_1,x_2\in[-\infty,-1]$$

$$\therefore$$ $$(1-x_2^2)(1-x_1^2)>0$$

$$\because$$ $$x_2>x_1$$ and $$x_1x_1$$

Thus,"$f(x_2)-f(x_1)=\frac{1}{1-x_2^2}-\frac{1}{1-x_1^2}=\frac{x_2^2-x_1^2}{(1-x_2^2)(1-x_1^2)}$"$$\because$$ both $$x_1,x_2\in[-\infty,-1]$$

$$\therefore$$$$(1-x_2^2)(1-x_1^2)>0$$

However, the sign of $$x_2^2-x_1^2$$ in $$(-1,1)$$ cannot be determined. It can only be determined in $$(-1,0)\text{ and }(0,1)$$. In $$(-1,0)$$

$$\because$$ $$x_2>x_1$$ and $$x_1x_1\text{ and }00$$ As a result,

$$f(x)$$ is decreasing in $$(-1,0)$$ and increasing in $$(0,1)$$. In $$(1,\infty)$$ Let $$x_1,x_2\in[1,\infty]$$ and $$x_2>x_1$$

Thus,"$f(x_2)-f(x_1)=\frac{1}{1-x_2^2}-\frac{1}{1-x_1^2}=\frac{x_2^2-x_1^2}{(1-x_2^2)(1-x_1^2)}$"$$\because$$ both $$x_1,x_2\in[-\infty,-1]$$

$$\therefore$$$$(1-x_2^2)(1-x_1^2)>0$$

$$\because x_2>x_1\text{ and }00$$"So, $f(x_2)-f(x_1)=\frac{x_2^2-x_1^2}{(1-x_2^2)(1-x_1^2)}>0$"$$f(x)$$ is increasing in $$(1,\infty)$$. Therefore, the intervals in which the function is increasing are $$(0,1)\And(1,\infty)$$.

$$\blacksquare$$

After learning derivatives, there will be more ways to determine the growth of a function.

Parity
The properties odd and even are associated with symmetry. While even functions have a symmetry about the $$y$$-axis, odd functions are symmetric about the origin. In mathematical terms: A function is even when $$f(-x)=f(x)$$

A function is odd when $$f(-x)=-f(x)$$ Example: Prove that $$f(x)=\frac{1}{1-x^2}$$ is an even function.

$$\because f(-x)=\frac{1}{1-(-x)^2}=\frac{1}{1-x^2}=f(x)$$

$$\therefore f(x) $$ is an even function

$$\blacksquare $$

Addition, Subtraction, Multiplication and Division of functions
For two real-valued functions, we can add the functions, multiply the functions, raised to a power, etc.

If we add the functions $$y=3x+2$$ and $$y=x^2$$, we obtain $$y=x^2+3x+2$$.

If we subtract $$y=3x+2$$ from $$y=x^2$$, we obtain $$y=x^2-(3x+2)$$. We can also write this as $$y=x^2-3x-2$$.

If we multiply the function $$y=3x+2$$ and the function $$y=x^2$$, we obtain $$y=(3x+2)x^2$$. We can also write this as $$y=3x^3+2x^2$$.

If we divide the function $$y=3x+2$$ by the function $$y=x^2$$, we obtain $$y=\frac{3x+2}{x^2}$$.

If a math problem wants you to add two functions $$f$$ and $$g$$, there are two ways that the problem will likely be worded:

 * 1) If you are told that $$f(x)=3x+2$$, that $$g(x)=x^2$$ , that $$h(x)=f(x)+g(x)$$ and asked about $$h$$ , then you are being asked to add two functions. Your answer would be $$h(x)=x^2+3x+2$$.
 * 2) If you are told that $$f(x)=3x+2$$, that $$g(x)=x^2$$ and you are asked about $$f+g$$ , then you are being asked to add two functions. The addition of $$f$$ and $$g$$ is called $$f+g$$ . Your answer would be $$(f+g)(x)=x^2+3x+2$$.

Similar statements can be made for subtraction, multiplication and division.

Let $$f(x)=3x+2$$ and: $$g(x)=x^2$$. Let's add, subtract, multiply and divide.

 * $$\begin{align}

(f+g)(x) &= f(x)+g(x)\\ &= (3x+2)+(x^2)\\ &= x^2+3x+2\, \end{align}$$ ,

 * $$\begin{align}

(f-g)(x) &= f(x)-g(x)\\ &= (3x+2)-(x^2)\\ &= -x^2+3x+2\, \end{align}$$ ,

 * $$\begin{align}

(f\times g)(x) &= f(x)\times g(x)\\ &= (3x+2)\times(x^2)\\ &= 3x^3+2x^2\, \end{align}$$ ,

 * $$\begin{align}

\left(\frac{f}{g}\right)(x) &= \frac{f(x)}{g(x)}\\ &= \frac{3x+2}{x^2}\\ &= \frac{3}{x}+\frac{2}{x^2} \end{align}$$.

Composition of functions
We begin with a fun (and not too complicated) application of composition of functions before we talk about what composition of functions is.

If we drop a ball from a bridge which is 20 meters above the ground, then the height of our ball above the earth is a function of time. The physicists tell us that if we measure time in seconds and distance in meters, then the formula for height in terms of time is $$h=-4.9t^2+20$$. Suppose we are tracking the ball with a camera and always want the ball to be in the center of our picture. Suppose we have $$\theta=f(h)$$ The angle will depend upon the height of the ball above the ground and the height above the ground depends upon time. So the angle will depend upon time. This can be written as $$\theta=f(-4.9t^2+20)$$. We replace $$h$$ with what it is equal to. This is the essence of composition.

Composition of functions is another way to combine functions which is different from addition, subtraction, multiplication or division.

The value of a function $$f$$ depends upon the value of another variable $$x$$ ; however, that variable could be equal to another function $$g$$, so its value depends on the value of a third variable. If this is the case, then the first variable is a function $$h$$ of the third variable; this function ($$h$$) is called the composition of the other two functions ($$f$$ and $$g$$).

Let $$f(x)=3x+2$$ and: $$g(x)=x^2$$. The composition of $$f$$ with $$g$$ is read as either "f composed with g" or "f of g of x."

Let
 * $$h(x)=f(g(x))$$

Then

 * $$\begin{align}

h(x) &= f(g(x))\\ &= f(x^2)\\ &= 3(x^2)+2\\ &= 3x^2+2\, \end{align}$$.

Sometimes a math problem asks you compute $$(f\circ g)(x)$$ when they want you to compute $$f(g(x))$$ ,

Here, $$h$$ is the composition of $$f$$ and $$g$$ and we write $$h=f\circ g$$. Note that composition is not commutative:

 * $$f(g(x))=3x^2+2$$, and

 * $$\begin{align}

g(f(x)) &= g(3x+2)\\ &= (3x+2)^2\\ &= 9x^2+12x+4\,. \end{align}$$

 * so $$f(g(x))\ne g(f(x))$$.

Composition of functions is very common, mainly because functions themselves are common. For instance, squaring and sine are both functions:

 * $$\mbox{square}(x)=x^2$$

 * $$\mbox{sine}(x)=\sin(x)$$

Thus, the expression $$\sin^2(x)$$ is a composition of functions:

 * $$\sin^2(x)$$
 * $$=\mbox{square}(\sin(x))$$
 * $$=\mbox{square}(\mbox{sine}(x))$$
 * }
 * $$=\mbox{square}(\mbox{sine}(x))$$
 * }
 * }

(Note that this is not the same as $$\mbox{sine}(\mbox{square}(x))=\sin(x^2)$$ .) Since the function sine equals $$\tfrac{1}{2}$$ if $$x=\frac{\pi}{6}$$ ,

 * $$\mbox{square}\left(\sin\left(\tfrac{\pi}{6}\right)\right)=\mbox{square}\left(\tfrac{1}{2}\right)$$.

Since the function square equals $$\frac{1}{4}$$ if $$x=\frac{\pi}{6}$$ ,

 * $$\sin^2\left(\tfrac{\pi}{6}\right)=\mbox{square}\left(\sin\left(\tfrac{\pi}{6}\right)\right)=\mbox{square}\left(\tfrac{1}{2}\right)=\frac{1}{4}$$.

Transformations
Transformations are a type of function manipulation that are very common. They consist of multiplying, dividing, adding or subtracting constants to either the input or the output. Multiplying by a constant is called dilation and adding a constant is called translation. Here are a few examples:

 * $$f(2\times x)$$ Dilation
 * $$f(x+2)$$ Translation
 * $$2\times f(x)$$ Dilation
 * $$2+f(x)$$ Translation

Translations and dilations can be either horizontal or vertical. Examples of both vertical and horizontal translations can be seen at right. The red graphs represent functions in their 'original' state, the solid blue graphs have been translated (shifted) horizontally, and the dashed graphs have been translated vertically.

Dilations are demonstrated in a similar fashion. The function
 * $$f(2\times x)$$

has had its input doubled. One way to think about this is that now any change in the input will be doubled. If I add one to $$x$$, I add two to the input of $$f$$, so it will now change twice as quickly. Thus, this is a horizontal dilation by $$\frac{1}{2}$$ because the distance to the $$y$$-axis has been halved. A vertical dilation, such as
 * $$2\times f(x)$$

is slightly more straightforward. In this case, you double the output of the function. The output represents the distance from the $$x$$-axis, so in effect, you have made the graph of the function 'taller'. Here are a few basic examples where $$a$$ is any positive constant:

Inverse functions
We call $$g(x)$$ the inverse function of $$f(x)$$ if, for all $$x$$ :
 * $$g(f(x))=f(g(x))=x$$

A function $$f(x)$$ has an inverse function if and only if $$f(x)$$ is one-to-one. For example, the inverse of $$f(x)=x+2$$ is $$g(x)=x-2$$. The function $$f(x)=\sqrt{1-x^2}$$ has no inverse because it is not injective. Similarly, the inverse functions of trigonometric functions have to undergo transformations and limitations to be considered as valid functions.

Notation
The inverse function of $$f$$ is denoted as $$f^{-1}(x)$$. Thus, $$f^{-1}(x)$$ is defined as the function that follows this rule
 * $$f(f^{-1}(x))=f^{-1}(f(x))=x$$

To determine $$f^{-1}(x)$$ when given a function $$f$$, substitute $$f^{-1}(x)$$ for $$x$$ and substitute $$x$$ for $$f(x)$$. Then solve for $$f^{-1}(x)$$, provided that it is also a function.

Example: Given $$f(x)=2x-7$$, find $$f^{-1}(x)$$.

Substitute $$f^{-1}(x)$$ for $$x$$ and substitute $$x$$ for $$f(x)$$. Then solve for $$f^{-1}(x)$$ :

 * $$f(x)=2x-7$$

 * $$\Leftrightarrow x=2[f^{-1}(x)]-7$$

 * $$\Leftrightarrow x+7=2[f^{-1}(x)]$$

 * $$\Leftrightarrow\frac{x+7}{2}=f^{-1}(x)$$

To check your work, confirm that $$f^{-1}(f(x))=x$$ :"$f^{-1}(f(x))=$$f^{-1}(2x-7)={}$$\frac{(2x-7)+7}{2}=\frac{2x}{2}=x$"If $$f$$ isn't one-to-one, then, as we said before, it doesn't have an inverse. Then this method will fail.

Example: Given $$f(x)=x^2$$, find $$f^{-1}(x)$$.

Substitute $$f^{-1}(x)$$ for $$x$$ and substitute $$x$$ for $$f(x)$$. Then solve for $$f^{-1}(x)$$ :

 * $$f(x)=x^2$$

 * $$x=(f^{-1}(x))^2$$

 * $$f^{-1}(x)=\pm\sqrt{x}$$

Since there are two possibilities for $$f^{-1}(x)$$, it's not a function. Thus $$f(x)=x^2$$ doesn't have an inverse. Of course, we could also have found this out from the graph by applying the Horizontal Line Test. It's useful, though, to have lots of ways to solve a problem, since in a specific case some of them might be very difficult while others might be easy. For example, we might only know an algebraic expression for $$f(x)$$ but not a graph.

=External links=
 * Videos on function and their graphs at Khan Academy

Analyse/Functies

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Calculus/Functions


text:
Waves/Vectors

Math Tutorial -- Vectors

Figure 1: Displacement vectors in a plane.

Vector $$\mbox{A}$$ represents the displacement of George from Mary, while vector $$\mbox{B}$$ represents the displacement of Paul from George. Vector $$\mbox{C}$$ represents the displacement of Paul from Mary and $$\mbox{C} = \mbox{A} + \mbox{B}$$. The quantities $$A_x$$, $$A_y$$, etc., represent the Cartesian components of the vectors.

Before we can proceed further we need to explore the idea of a vector. A vector is a quantity which expresses both magnitude and direction. Graphically we represent a vector as an arrow. In typeset notation a vector is represented by a boldface character, while in handwriting an arrow is drawn over the character representing the vector.

Figure 1 shows some examples of displacement vectors, i. e., vectors which represent the displacement of one object from another, and introduces the idea of vector addition. The tail of vector $$\mbox{B}$$ is collocated with the head of vector $$\mbox{A}$$, and the vector which stretches from the tail of $$\mbox{A}$$ to the head of $$\mbox{B}$$ is the sum of $$\mbox{A}$$ and $$\mbox{B}$$, called $$\mbox{C}$$ in figure 1.

Figure 2: Definition sketch for the angle $$\theta $$ representing the orientation of a two dimensional vector.

The quantities $$A_x$$, $$A_y$$, etc., represent the Cartesian components of the vectors in figure 2. A vector can be represented either by its Cartesian components, which are just the projections of the vector onto the Cartesian coordinate axes, or by its direction and magnitude. The direction of a vector in two dimensions is generally represented by the counterclockwise angle of the vector relative to the $$x$$ axis, as shown in figure 2. Conversion from one form to the other is given by the equations

 * $$ A = (A_x^2 + A_y^2)^{1/2} \qquad \theta = \tan^{-1} (A_y /A_x ), $$

 * $$ A_x = A \cos ( \theta ) \qquad A_y = A \sin ( \theta ), $$

where $$A$$ is the magnitude of the vector. A vector magnitude is sometimes represented by absolute value notation: $$A \equiv \vert \mbox{A} \vert$$.

Notice that the inverse tangent gives a result which is ambiguous relative to adding or subtracting integer multiples of $$\pi$$. Thus the quadrant in which the angle lies must be resolved by independently examining the signs of $$A_x$$ and $$A_y$$ and choosing the appropriate value of $$\theta $$.

To add two vectors, $$\mbox{A}$$ and $$\mbox{B}$$, it is easiest to convert them to Cartesian component form. The components of the sum $$\mbox{C} = \mbox{A} + \mbox{B}$$ are then just the sums of the components:

 * $$ C_x = A_x + B_x \qquad C_y = A_y + B_y . $$

Subtraction of vectors is done similarly, e. g., if $$\mbox{A} = \mbox{C} - \mbox{B}$$, then

 * $$ A_x = C_x - B_x \qquad A_y = C_y - B_y . $$

A unit vector is a vector of unit length. One can always construct a unit vector from an ordinary vector by dividing the vector by its length: $$\mbox{n} = \mbox{A} / \vert \mbox{A} \vert$$. This division operation is carried out by dividing each of the vector components by the number in the denominator. Alternatively, if the vector is expressed in terms of length and direction, the magnitude of the vector is divided by the denominator and the direction is unchanged.

Unit vectors can be used to define a Cartesian coordinate system. Conventionally, $$\mbox{i}$$, $$\mbox{j}$$, and $$\mbox{k}$$ indicate the $$x$$, $$y$$, and $$z$$ axes of such a system. Note that $$\mbox{i}$$, $$\mbox{j}$$, and $$\mbox{k}$$ are mutually perpendicular. Any vector can be represented in terms of unit vectors and its Cartesian components: $$\mbox{A} = A_x \mbox{i} + A_y \mbox{j} + A_z \mbox{k}$$. An alternate way to represent a vector is as list of components: $$\mbox{A} = ( A_x, A_y , A_z )$$. We tend to use the latter representation since it is somewhat more economical notation.

There are two ways to multiply two vectors, yielding respectively what are known as the dot product and the cross product. The cross product yields another vector while the dot product yields a number. Here we will discuss only the dot product.

Figure 3: Definition sketch for dot product.

Given vectors $$\mbox{A}$$ and $$\mbox{B}$$, the dot product of the two is defined

 * $$ \mbox{A} \cdot \mbox{B} \equiv \vert \mbox{A} \vert \vert \mbox{B} \vert \cos \theta, $$

where $$\theta $$ is the angle between the two vectors. An alternate expression for the dot product exists in terms of the Cartesian components of the vectors:

 * $$ \mbox{A} \cdot \mbox{B} = A_x B_x + A_y B_y . $$

It is easy to show that this is equivalent to the cosine form of the dot product when the $$x$$ axis lies along one of the vectors, as in figure 3. Notice in particular that $$A_x = \vert \mbox{A} \vert \cos \theta$$, while $$B_x = \vert \mbox{B} \vert$$ and $$B_y = 0$$. Thus, $$\mbox{A} \cdot \mbox{B} = \vert \mbox{A} \vert \cos \theta \vert \mbox{B} \vert$$ in this case, which is identical to the form given above.

By the law of cosines we can also see that

 * $$\mathbf{A} \cdot \mathbf{B} = \frac{1}{2} \left(

 * \mathbf{A}+\mathbf{B}|^2-|\mathbf{A}|^2-|\mathbf{B}|^2 \right)$$

which is an alternate coordinate-free expression for the dot product.

Figure 4: Definition figure for rotated coordinate system. The vector $$\mbox{R}$$ has components $$X$$ and $$Y$$ in the unprimed coordinate system and components $$X'$$ and $$Y'$$ in the primed coordinate system.

All that remains to be proven for equation (2.6) to hold in general is to show that it yields the same answer regardless of how the Cartesian coordinate system is oriented relative to the vectors. To do this, we must show that $$A_x B_x + A_y B_y = A_x' B_x' + A_y' B_y'$$, where the primes indicate components in a coordinate system rotated from the original coordinate system.

This can be shown nearly instantly by applying the pythagorean theorem. Due to the fact that R is invariant and represents the hypotenuse for both triangles (X, X', Y and Y') we can conclude:

$$|R|^2 = X^2 + Y^2 \mbox{ and } X^2 + Y^2 = X'^2 + Y'^2$$
 * R|^2 = X'^2 + Y'^2 \Rightarrow

Since the dot product can be written solely in terms of magnitudes, as we did above, if the magnitude of a vector is invariant the dot product of two vectors must also be invariant.

To deduce a general formula for X' and Y' you will have to do a bit more thinking:

Figure 2.4 shows the vector $$\mbox{R}$$ resolved in two coordinate systems rotated with respect to each other. From this figure it is clear that $$X' = A + B$$. Focusing on the shaded triangles, we see that $$A = X \cos \theta$$ and $$B = Y \sin \theta$$. Thus, we find $$X' = X \cos \theta + Y \sin \theta$$. Similar reasoning shows that $$Y' = -X \sin \theta + Y \cos \theta$$ (Just imagine to rotate the constructs in the image further 90° without changing the axis-names. You will instantly notice that in the second quadrant X is negative while Y positive).

Thus, the new and old coordinates are related by

\begin{pmatrix} X^\prime \\ Y^\prime \end{pmatrix} = \begin{pmatrix} X \cos \theta + Y \sin \theta \\ -X \sin \theta + Y \cos \theta \end{pmatrix}$$

This is true of the position vector. We can use it to extend the notion of vector to concepts other than position by stating that a pair of numbers is a vector if and only if its values change in exactly this way under rotation.

Substituting this relation into our earlier expression for the dot product and using the trigonometric identity $$\cos^2 \theta + \sin^2 \theta = 1$$ results in

 * $$\begin{matrix}

A_x' B_x' + A_y' B_y' & = &  & (A_x \cos \theta + A_y \sin \theta ) (B_x \cos \theta + B_y \sin \theta ) \\ & & + & (-A_x \sin \theta + A_y \cos \theta ) (-B_x \sin \theta + B_y \cos \theta ) \\ & = & & A_x B_x + A_y B_y \end{matrix}$$

which proves the complete equivalence of the two forms of the dot product quoted above. (Multiply out the above expression to verify this.)

A numerical quantity which doesn't depend on which coordinate system is being used is called a scalar. The dot product of two vectors is a scalar. However, the components of a vector, taken individually, are not scalars, since the components change as the coordinate system changes. Since the laws of physics cannot depend on the choice of coordinate system being used, we insist that physical laws be expressed in terms of scalars and vectors, but not in terms of the components of vectors.

In three dimensions the cosine form of the dot product remains the same, while the component form is
 * $$ \mbox{A} \cdot \mbox{B} = A_x B_x + A_y B_y + A_z B_z . $$

title: Waves/Vectors


text:
Waves/Reflection and Refraction

Reflection and Refraction
Most of what we need to know about geometrical optics can be summarized in two rules, the laws of reflection and refraction. These rules may both be inferred by considering what happens when a plane wave segment impinges on a flat surface. If the surface is polished metal, the wave is reflected, whereas if the surface is an interface between two transparent media with differing indices of refraction, the wave is partially reflected and partially refracted. Reflection means that the wave is turned back into the half-space from which it came, while refraction means that it passes through the interface, acquiring a different direction of motion from that which it had before reaching the interface.

Figure 3.1 shows the wave vector and wave front of a wave being reflected from a plane mirror. The angles of incidence, $$\theta _I$$, and reflection, $$\theta _R$$, are defined to be the angles between the incoming and outgoing wave vectors respectively and the line normal to the mirror. The law of reflection states that $$\theta_R = \theta_I$$. This is a consequence of the need for the incoming and outgoing wave fronts to be in phase with each other all along the mirror surface. This plus the equality of the incoming and outgoing wavelengths is sufficient to insure the above result.

Refraction, as illustrated in figure 3.2, is slightly more complicated. Since $$n_R > n_I$$, the speed of light in the right-hand medium is less than in the left-hand medium. (Recall that the speed of light in a medium with refractive index $$n$$ is $$c_{medium} = c_{vac}/n$$.) The frequency of the wave packet doesn't change as it passes through the interface, so the wavelength of the light on the right side is less than the wavelength on the left side.

Let us examine the triangle ABC in figure 3.2. The side AC is equal to the side BC times $$\sin ( \theta_I )$$. However, AC is also equal to $$2 \lambda_I$$, or twice the wavelength of the wave to the left of the interface. Similar reasoning shows that $$2 \lambda_R$$, twice the wavelength to the right of the interface, equals BC times $$\sin ( \theta_R )$$. Since the interval BC is common to both triangles, we easily see that
 * $$ \frac{\lambda_I}{\lambda_R} = \frac{\sin ( \theta_I )}{\sin ( \theta_R )} . $$ (4.1)

Since $$\lambda_I = c_I T = c_{vac} T/n_I$$ and $$\lambda_R = c_R T = c_{vac} T/n_R$$ where $$c_I$$ and $$c_R$$ are the wave speeds to the left and right of the interface, $$c_{vac}$$ is the speed of light in a vacuum, and $$T$$ is the (common) period, we can easily recast the above equation in the form
 * $$ n_I \sin ( \theta_I ) = n_R \sin ( \theta_R ) . $$ (4.2)

This is called Snell's law, and it governs how a ray of light bends as it passes through a discontinuity in the index of refraction. The angle $$\theta _I$$ is called the incident angle and $$\theta _R$$ is called the refracted angle. Notice that these angles are measured from the normal to the surface, not the tangent.

Derivation for Law of Reflection
The derivation of Law of Reflection using Fermat's principle is straightforward. The Law of Reflection can be derived using elementary Calculus and Trigonometry. The generalization of the Law of Reflection is Snell's law, which is derived bellow using the same principle.

The medium that light travels through doesn't change. In order to minimize the time for light travel between to points, we should minimize the path taken.

1. Total path length of the light is given by

 * $$L=d_1+d_2\,$$

2. Using Pythagorean theorem from Euclidean Geometry we see that

 * $$d_1=\sqrt{x^2 + a^2}\,$$ and $$d_2=\sqrt{(l-x)^2 + b^2}\,$$

3. When we substitute both values of d1 and d2 for above, we get

 * $$L=\sqrt{x^2 + a^2} + \sqrt{(l-x)^2 + b^2}$$

4. In order to minimize the path traveled by light, we take the first derivative of L with respect to x.

 * $$\frac{dL}{dx}=\frac{x}{\sqrt{x^2 + a^2}} + \frac{-(l-x)}{\sqrt{(l-x)^2 + b^2}}=0$$

5. Set both sides equal to each other.

 * $$\frac{x}{\sqrt{x^2 + a^2}} = \frac{(l-x)}{\sqrt{(l-x)^2 + b^2}}$$

6. We can now tell that the left side is nothing but $$\sin\theta_i$$ and the right side $$\sin\theta_r$$ means

 * $$\sin\theta_i=\sin\theta_r \!\ $$

7. Taking the inverse sine of both sides we see that the angle of incidence equals the angle of reflection

 * $$\theta_i=\theta_r \!\ $$

Derivation for Snell's Law
The derivation of Snell's Law using Fermat's Priciple is straightforward. Snell's Law can be derived using elementary calculus and trigonometry. Snell's Law is the generalization of the above in that it does not require the medium to be the same everywhere.

To mark the speed of light in different media refractive indices named n1 and n2 are used.
 * $$v_1=\frac{c}{n_1} \!\ $$
 * $$v_2=\frac{c}{n_2}$$

Here $$c$$ is the speed of light in the vacuum and $$n_1,n_2 \ge 1 \,$$ because all materials slow down light as it travels through them.

1. Time for the trip equals distance traveled divided by the speed.

 * $$t=\frac{d_1}{v_1}+\frac{d_2}{v_2}$$

2. Using the Pythagorean theorem from Euclidean Geometry we see that

 * $$\frac{d_1}{v_1}=\frac{\sqrt{x^2 + a^2}}{v_1}\,$$ and $$\frac{d_2}{v_2}=\frac{\sqrt{b^2 + (l-x)^2}}{v_2}\,$$

3. Substituting this result into equation (1) we get

 * $$t=\frac{\sqrt{x^2 + a^2}}{v_1} + \frac{\sqrt{b^2 + (l-x)^2}}{v_2}$$

4. Differentiating and setting the derivative equal to zero gives

 * $$\frac{dt}{dx}=\frac{x}{v_1\sqrt{x^2 + a^2}} + \frac{-(l-x)}{v_2\sqrt{(l-x)^2 + b^2}}=0$$

5. After careful examination the above equation we see that it is nothing but

 * $$\frac{dt}{dx}=\frac{\sin\theta_1}{v_1} - \frac{\sin\theta_2}{v_2}=0$$

6. Thus

 * $$\frac{v_1}{\sin\theta_1}=\frac{v_2}{\sin\theta_2}$$

7. Multiplying both sides by $$\sin\theta_1\sin\theta_2$$ we get

 * $$v_1\sin\theta_2=v_2\sin\theta_1 \!\ $$

8. Substituting $$\frac{c}{n_1}$$ for v1 and $$\frac{c}{n_2}$$ for $$v_2$$ we get

 * $$\frac{c}{n_1}\sin\theta_2=\frac{c}{n_2}\sin\theta_1$$

9. Simplifying both sides we get our final result
 * $$n_1\sin\theta_1=n_2\sin\theta_2 \!\ $$

title: Waves/Reflection and Refraction


text:
Special Relativity/Spacetime

The modern approach to relativity
Although the special theory of relativity was first proposed by Einstein in 1905, the modern approach to the theory depends upon the concept of a four-dimensional universe, that was first proposed by Hermann Minkowski in 1908.

Minkowski's contribution appears complicated but is simply an extension of Pythagoras' Theorem:

In 2 dimensions: $$h^2 = x^2 + y^2$$

In 3 dimensions: $$h^2 = x^2 + y^2 + z^2$$

in 4 dimensions: $$s^2 = x^2 + y^2 + z^2 + kt^2$$

(where k = $$-c^2$$)

The modern approach uses the concept of invariance to explore the types of coordinate systems that are required to provide a full physical description of the location and extent of things. The modern theory of special relativity begins with the concept of "length". In everyday experience, it seems that the length of objects remains the same no matter how they are rotated or moved from place to place. We think that the simple length of a thing is "invariant". However, as is shown in the illustrations below, what we are actually suggesting is that length seems to be invariant in a three-dimensional coordinate system.

The length of a thing in a two-dimensional coordinate system is given by Pythagoras's theorem:

 * $$ x^2 + y^2 = h^2 $$

This two-dimensional length is not invariant if the thing is tilted out of the two-dimensional plane. In everyday life, a three-dimensional coordinate system seems to describe the length fully. The length is given by the three-dimensional version of Pythagoras's theorem:

 * $$ h^2 = x^2 + y^2 + z^2 $$

The derivation of this formula is shown in the illustration below.

It seems that, provided all the directions in which a thing can be tilted or arranged are represented within a coordinate system, then the coordinate system can fully represent the length of a thing. However, it is clear that things may also be changed over a period of time. Time is another direction in which things can be arranged. This is shown in the following diagram:

The length of a straight line between two events in space and time is called a "space-time interval".

In 1908 Hermann Minkowski pointed out that if things could be rearranged in time, then the universe might be four-dimensional. He boldly suggested that Einstein's recently-discovered theory of Special Relativity was a consequence of this four-dimensional universe. He proposed that the space-time interval might be related to space and time by Pythagoras' theorem in four dimensions:

 * $$ s^2 = x^2 + y^2 + z^2 + (ict)^2 $$

Where i is the imaginary unit (sometimes imprecisely called $$\sqrt{-1}$$), c is a constant, and t is the time interval spanned by the space-time interval, s. The symbols x, y and z represent displacements in space along the corresponding axes. In this equation, the 'second' becomes just another unit of length. In the same way as centimetres and inches are both units of length related by centimetres = 'conversion constant' times inches, metres and seconds are related by metres = 'conversion constant' times seconds. The conversion constant, c has a value of about 300,000,000 meters per second. Now $$i^2$$ is equal to minus one, so the space-time interval is given by:

 * $$ s^2 = x^2 + y^2 + z^2 - (ct)^2 $$

Minkowski's use of the imaginary unit has been superseded by the use of advanced geometry that uses a tool known as the "metric tensor". The metric tensor permits the existence of "real" time and the negative sign in the expression for the square of the space-time interval originates in the way that distance changes with time when the curvature of spacetime is analysed (see advanced text). We now use real time but Minkowski's original equation for the square of the interval survives so that the space-time interval is still given by:

 * $$ s^2 = x^2 + y^2 + z^2 - (ct)^2 $$

Space-time intervals are difficult to imagine; they extend between one place and time and another place and time, so the velocity of the thing that travels along the interval is already determined for a given observer.

If the universe is four-dimensional, then the space-time interval (rather than the spatial length) will be invariant. Whoever measures a particular space-time interval will get the same value, no matter how fast they are travelling. In physical terminology the invariance of the spacetime interval is a type of Lorentz Invariance. The invariance of the spacetime interval has some dramatic consequences.

The first consequence is the prediction that if a thing is travelling at a velocity of c metres per second, then all observers, no matter how fast they are travelling, will measure the same velocity for the thing. The velocity c will be a universal constant. This is explained below.

When an object is travelling at c, the space time interval is zero, this is shown below:

 * The distance travelled by an object moving at velocity v in the x direction for t seconds is:

 * $$ x = vt $$

 * If there is no motion in the y or z directions the space-time interval is $$ s^2 = x^2 + 0 + 0 - (ct)^2 $$

 * So: $$ s^2 = (vt)^2 - (ct)^2 $$

 * But when the velocity v equals c:

 * $$ s^2 = (ct)^2 - (ct)^2 $$

 * And hence the space time interval $$ s^2 = (ct)^2 - (ct)^2 = 0 $$

A space-time interval of zero only occurs when the velocity is c (if x>0). All observers observe the same space-time interval so when observers observe something with a space-time interval of zero, they all observe it to have a velocity of c, no matter how fast they are moving themselves.

The universal constant, c, is known for historical reasons as the "speed of light in a vacuum". In the first decade or two after the formulation of Minkowski's approach many physicists, although supporting Special Relativity, expected that light might not travel at exactly c, but might travel at very nearly c. There are now few physicists who believe that light in a vacuum does not propagate at c.

The second consequence of the invariance of the space-time interval is that clocks will appear to go slower on objects that are moving relative to you. Suppose there are two people, Bill and John, on separate planets that are moving away from each other. John draws a graph of Bill's motion through space and time. This is shown in the illustration below:

Being on planets, both Bill and John think they are stationary, and just moving through time. John spots that Bill is moving through what John calls space, as well as time, when Bill thinks he is moving through time alone. Bill would also draw the same conclusion about John's motion. To John, it is as if Bill's time axis is leaning over in the direction of travel and to Bill, it is as if John's time axis leans over.

 * John calculates the length of Bill's space-time interval as:

 * $$s^2 = (vt)^2 - (ct)^2$$

 * whereas Bill doesn't think he has travelled in space, so writes:

 * $$s^2 = (0)^2 - (cT)^2$$

The space-time interval, $$s^2$$, is invariant. It has the same value for all observers, no matter who measures it or how they are moving in a straight line. Bill's $$ s^2$$ equals John's $$ s^2$$ so:

 * $$(0)^2 - (cT)^2 = (vt)^2 - (ct)^2$$

 * and

 * $$-(cT)^2 = (vt)^2 - (ct)^2$$

 * hence

 * $$t = T / \sqrt{1 - v^2/c^2}$$.

So, if John sees Bill measure a time interval of 1 second ($$T = 1$$) between two ticks of a clock that is at rest in Bill's frame, John will find that his own clock measures between these same ticks an interval $$t$$, called coordinate time, which is greater than one second. It is said that clocks in motion slow down, relative to those on observers at rest. This is known as "relativistic time dilation of a moving clock". The time that is measured in the rest frame of the clock (in Bill's frame) is called the proper time of the clock.

John will also observe measuring rods at rest on Bill's planet to be shorter than his own measuring rods, in the direction of motion. This is a prediction known as "relativistic length contraction of a moving rod". If the length of a rod at rest on Bill's planet is $$X$$, then we call this quantity the proper length of the rod. The length $$x$$ of that same rod as measured from John's planet, is called coordinate length, and given by

 * $$x = X \sqrt{1 - v^2/c^2}$$.

This equation can be derived directly and validly from the time dilation result with the assumption that the speed of light is constant.

The last consequence is that clocks will appear to be out of phase with each other along the length of a moving object. This means that if one observer sets up a line of clocks that are all synchronised so they all read the same time, then another observer who is moving along the line at high speed will see the clocks all reading different times. In other words observers who are moving relative to each other see different events as simultaneous. This effect is known as Relativistic Phase or the Relativity of Simultaneity. Relativistic phase is often overlooked by students of Special Relativity, but if it is understood then phenomena such as the twin paradox are easier to understand.

The way that clocks go out of phase along the line of travel can be calculated from the concepts of the invariance of the space-time interval and length contraction.

In the diagram above John is conventionally stationary. Distances between two points according to Bill are simple lengths in space (x) all at t=0 whereas John sees Bill's measurement of distance as a combination of a distance (X) and a time interval (T):

 * $$x^2 = X^2 - (cT)^2$$

Notice that the quantities represented by capital letters are proper lengths and times and in this example refer to John's measurements.

Bill's distance, x, is the length that he would obtain for things that John believes to be X metres in length. For Bill it is John who has rods that contract in the direction of motion so Bill's determination "x" of John's distance "X" is given from:

 * $$x = X \sqrt{1 - v^2/c^2}$$.

This relationship between proper and coordinate lengths was seen above to relate Bill's proper lengths to John's measurements. It also applies to how Bill observes John's proper lengths.

 * $$x = X \sqrt{1 - v^2/c^2}$$

 * Thus $$x^2 = X^2 - (v^2/c^2)X^2$$

 * So: $$(cT)^2 = (v^2/c^2)X^2$$

 * And $$cT = (v/c)X$$

 * So: $$T = (v/c^2)X $$

Clocks that are synchronised for one observer go out of phase along the line of travel for another observer moving at $$v$$ metres per second by :$$(v/c^2)$$ seconds for every metre. This is one of the most important results of Special Relativity and should be thoroughly understood by students.

The net effect of the four-dimensional universe is that observers who are in motion relative to you seem to have time coordinates that lean over in the direction of motion and consider things to be simultaneous that are not simultaneous for you. Spatial lengths in the direction of travel are shortened, because they tip upwards and downwards, relative to the time axis in the direction of travel, akin to a rotation out of three-dimensional space.

 * Examp.jpgExample

 * An observer records an event A next to herself then an event B a millisecond later and 600 kilometres away. How fast would another observer need to travel in the direction of event B to record it as simultaneous with event A?

 * The phase difference between moving clocks is given by: $$T = (v/c^2)X $$
 * The velocity needed to create a phase difference of 10-3 secs is:

 * $$v = \frac{10^{-3} \times 3 \times 10^8 \times c}{6 \times 10^{5}}$$

 * This means that another observer who is travelling at 0.5c relative to the first observer in the direction of the line joining events A and B will consider that these events are simultaneous.

Interpreting space-time diagrams
Great care is needed when interpreting space-time diagrams. Diagrams present data in two dimensions, and cannot show faithfully how, for instance, a zero length space-time interval appears.

When diagrams are used to show both space and time it is important to be alert to space and time being related by Minkowski's equation and not by simple Euclidean geometry. The diagrams are only aids to understanding the approximate relation between space and time and it must not be assumed, for instance, that simple trigonometric relationships can be used to relate lines that represent spatial displacements and lines that represent temporal displacements.

It is sometimes mistakenly held that the time dilation and length contraction results only apply for observers at x=0 and t=0. This is untrue. An inertial frame of reference is defined so that length and time comparisons can be made anywhere within a given reference frame.

Time differences in one inertial reference frame can be compared with time differences anywhere in another inertial reference frame provided it is remembered that these differences apply to corresponding pairs of lines or pairs of planes of simultaneous events.

Spacetime

In order to gain an understanding of both Galilean and Special Relativity it is important to begin thinking of space and time as being different dimensions of a four-dimensional vector space called spacetime. Actually, since we can't visualize four dimensions very well, it is easiest to start with only one space dimension and the time dimension. The figure shows a graph with time plotted on the vertical axis and the one space dimension plotted on the horizontal axis. An event is something that occurs at a particular time and a particular point in space. ("Julius X. wrecks his car in Lemitar, NM on 21 June at 6:17 PM.") A world line is a plot of the position of some object as a function of time (more properly, the time of the object as a function of position) on a spacetime diagram. Thus, a world line is really a line in spacetime, while an event is a point in spacetime. A horizontal line parallel to the position axis (x-axis) is a line of simultaneity; in Galilean Relativity all events on this line occur simultaneously for all observers. It will be seen that the line of simultaneity differs between Galilean and Special Relativity; in Special Relativity the line of simultaneity depends on the state of motion of the observer.

In a spacetime diagram the slope of a world line has a special meaning. Notice that a vertical world line means that the object it represents does not move -- the velocity is zero. If the object moves to the right, then the world line tilts to the right, and the faster it moves, the more the world line tilts. Quantitatively, we say that
 * $$velocity = \frac{1}{slope~of~world~line} . $$(5.1)

Notice that this works for negative slopes and velocities as well as positive ones. If the object changes its velocity with time, then the world line is curved, and the instantaneous velocity at any time is the inverse of the slope of the tangent to the world line at that time.

The hardest thing to realize about spacetime diagrams is that they represent the past, present, and future all in one diagram. Thus, spacetime diagrams don't change with time -- the evolution of physical systems is represented by looking at successive horizontal slices in the diagram at successive times. Spacetime diagrams represent the evolution of events, but they don't evolve themselves.

The lightcone
Things that move at the speed of light in our four dimensional universe have surprising properties. If something travels at the speed of light along the x-axis and covers x meters from the origin in t seconds the space-time interval of its path is zero.

$$s^2 = x^2 - (ct)^2$$

but $$ x = ct$$ so:

$$s^2 = (ct)^2 - (ct)^2 = 0 $$

Extending this result to the general case, if something travels at the speed of light in any direction into or out from the origin it has a space-time interval of 0:

$$0 = x^2 + y^2 + z^2 - (ct)^2 $$

This equation is known as the Minkowski Light Cone Equation. If light were travelling towards the origin then the Light Cone Equation would describe the position and time of emission of all those photons that could be at the origin at a particular instant. If light were travelling away from the origin the equation would describe the position of the photons emitted at a particular instant at any future time 't'.

At the superficial level the light cone is easy to interpret. Its backward surface represents the path of light rays that strike a point observer at an instant and its forward surface represents the possible paths of rays emitted from the point observer. Things that travel along the surface of the light cone are said to be light- like and the path taken by such things is known as a null geodesic.

Events that lie outside the cones are said to be space-like or, better still space separated because their space time interval from the observer has the same sign as space (positive according to the convention used here). Events that lie within the cones are said to be time-like or time separated because their space-time interval has the same sign as time.

However, there is more to the light cone than the propagation of light. If the added assumption is made that the speed of light is the maximum possible velocity then events that are space separated cannot affect the observer directly. Events within the backward cone can have affected the observer so the backward cone is known as the "affective past" and the observer can affect events in the forward cone hence the forward cone is known as the "affective future".

The assumption that the speed of light is the maximum velocity for all communications is neither inherent in nor required by four dimensional geometry although the speed of light is indeed the maximum velocity for objects if the principle of causality is to be preserved by physical theories (ie: that causes precede effects).

The Lorentz transformation equations
The discussion so far has involved the comparison of interval measurements (time intervals and space intervals) between two observers. The observers might also want to compare more general sorts of measurement such as the time and position of a single event that is recorded by both of them. The equations that describe how each observer describes the other's recordings in this circumstance are known as the Lorentz Transformation Equations. (Note that the symbols below signify coordinates.)

The table below shows the Lorentz Transformation Equations.

See mathematical derivation of Lorentz transformation.

Notice how the phase ( (v/c2)x ) is important and how these formulae for absolute time and position of a joint event differ from the formulae for intervals.

A spacetime representation of the Lorentz Transformation
Bill and John are moving at a relative velocity, v, and synchronise clocks when they pass each other. Both Bill and John observe an event along Bill's direction of motion. What times will Bill and John assign to the event? It was shown above that the relativistic phase was given by: $$vx/c^2$$. This means that Bill will observe an extra amount of time elapsing on John's time axis due to the position of the event. Taking phase into account and using the time dilation equation Bill is going to observe that the amount of time his own clocks measure can be compared with John's clocks using:

$$T = \frac {t - vx/c^2}{\sqrt{1 - v^2/c^2}}$$.

This relationship between the times of a common event between reference frames is known as the Lorentz Transformation Equation for time.

Continue

title: Special Relativity/Spacetime


text:
Talk:Waves/Waves in One Dimension

This page is missing a graph. "Graphing this out we get [graph] From figure one we can" There should be a graph where I placed the word [graph]. MicScoTho

I think that

 * $$y=a\ \sin \left(\frac{2\pi x}{\lambda} - \frac{2\pi t}{T} +\alpha \right)$$

is neater. It also allows to immediately identify that λ is the spatial period, and that its inverse, k, together with the inverse of T, f, are the spatial and temporal frequencies, respectively.

Along the same line the last equation (second line) in the page could read

 * $$ 0=\frac{2\pi}{\lambda}\left(x - \frac{\lambda}{T} t\right) +\alpha$$

It direcly shows that the equiphase point moves with velocity c=λ/T

RDR

Lets break this page up into smaller subunits so it is quicker to load and easier to navigate, etc. Keep up the great work, -- Karl Wick

I'm just trying to wikify the source work, and it makes more sense to me to do each chapter as a page. I should do another few chapters tomorrow, and hope to be finished by Monday. After that, I'm going to leave it alone; I didn't do well enough at physics in school to edit this! -- Jimregan 02:15 20 Jul 2003 (UTC)

To clarify, the chapters listed are done so far as I'm concerned; I've done what I can with them, that is, unless someone can show me how to fix the ugly formatting as regards the image layout, the numbering of the problems &c, in which case I'll gladly fix what I've done. -- Jimregan 02:32 20 Jul 2003 (UTC)

I was reading the introduction, and I know from my textbooks that the electromagnetic and the weak nuclear force are considered as one force called the electroweak force. So there are only three fundamental forces. Just trying to help.

Actually, there are four fundamental forces: gravity, electromagnetic, weak nuclear and strong nuclear. The electromagnetic and weak nuclear forces are unified, as electroweak, at energies greater than around 100 GeV. Many grand unified theories (GUTs) merge the electroweak with the strong nuclear force at higher energies. 27 Feb 2004

 * Just a note: GeV = gigaElectronVolts. Also: electricity and magnestism are two forces but because they unify at every energy level they are considered one.  Correct me if I'm wrong ... Dolphinn!! 9:02 EST April 6 2007

Merger
Waves/Sine Waves has some stuff about angular velocity, but is otherwise very similar to this page. I recommend that useful content from there be merged onto this page, with all the other math stuff. 170.140.237.114 (discuss) 14:20, 16 March 2011 (UTC)

I would agree, there really is not a need to have them in separate pages Rahulsanjay18 (discuss • contribs) 17:05, 8 October 2020 (UTC)

title: Talk:Waves/Waves in One Dimension


text:
User:Eric119

"...we who teach will be judged more strictly."

 * --James 3:1b (NIV)

w:User:Eric119

title: User:Eric119


text:
Talk:Waves/Reflection and Refraction

Lets break this up a bit so it isn't so long .. -- Karl Wick

Total internal Reflection
Sir, I would like to know in a physics point of view,keeping formulas & mathematics aside why total internal reflection takes place.

RSVP to : adityavalluri@tatanova.com

title: Talk:Waves/Reflection and Refraction


text:
User:LouI~enwikibooks

My normal use name is Lou Imholt. I'm a semi-retired computer nerd (system engineer) with 35+ years experience, and an apprentice nanotechnologist starting in December, 2002.

I am still wraping up some loose ends I've left Wikipedia, but I have started to contribute ober here. I expect most effort to go into a new start up venture, the Survey of Computing.

Use my talk page, or send an e-mail for comments, Thanks.

title: User:LouI~enwikibooks


text:
User talk:LouI~enwikibooks

Hey LouI, any work you contribute will be greatly appreciated. I think this thing will change education. -- Karl Wick

To whom it may concern:

I'm going to create a sandbox attached to me user page to try an experiment with pop up windows. It works in my browsers off-line but I need to try it within Wiki file structure. If it works, this would be useful in textbooks for glossary or footnote entries. I may wind up creating sub files for my sandbox, but I'll ask for it all to be removed after the trial.

Thanks ... LouI 04:46, 10 Aug 2003 (UTC)

Hi :) I've just noticed you changed the Study Guide:Shakespeare link to have a space inbetween the colon - it's not really common practice for namespaces to do so, (eg User talk:, Wikibooks:...)

Thanks :) Dysprosia 08:05, 2 Nov 2003 (UTC)

I've just seen the discussion on Staff lounge, never mind. Dysprosia 05:22, 5 Nov 2003 (UTC)

Loul...are you still interested in the Wiki-University thread?
I'm a new PhD student in Instructional technology, and I want to do my dissertation on open textbooks.

Can you give me an update on where things stand. did they die?

Thanks!

John

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called LouI. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name LouI~enwikibooks that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:26, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 05:12, 19 April 2015 (UTC)

title: User talk:LouI~enwikibooks


text:
Talk:Organic Chemistry/Cover

Splash page
I decided to make this page a "splash page" or a sort of bookcover. Just the title, welcome comment, link to the contents and a pretty, ochem-related piece of artwork. --Karl Wick
 * What a good idea! I'm going to do the same Theresa knott 09:16, 24 Sep 2003 (UTC)

title: Talk:Organic Chemistry/Cover


text:
User talk:Cyp

en:User talk:Cyp

Hej/Miav/Hello/Meow/Aloha/?/Oi/Miado

Hey :) Thanks for picking up my arithmetic errors; my mental arithmetic is atrocious! Dysprosia 21:10, 29 Sep 2003 (UTC)

Hi, nice job with all the puzzles! I'll see if I can find some more good and challenging ones. Btw: previously we first posted solution ideas to the talk page of a puzzle, but maybe it's better the way you did it and post directly to the solution page (incorrect / unclear solutions can always be edited later I guess). Thomas

 * Thanks. Only saw the talk pages with solution ideas, after making all the solution pages... Think the solutions are probably correct, although the elastic one may not exactly have been the intended solution... Cyp 18:01, 18 Oct 2003 (UTC)

Hello, the changes you made to the recursion section of primes is incorrect. Maybe i didn't explain recursion properly, and i'll try again. Xiaodai

 * I'm suppose I'm not actually sure what the definition of recursion is exactly... But I don't understand what the difference between iteration, and the example given, is. كсηפ Cyp 12:50, 26 Oct 2003 (UTC)

 * To define recursion, you first have to define recursion :) Thomas

-- From the RfA page:
 * Cyp - Oh well, why not... Not that I've actually had any need for admin stuff in the Wikibooks section yet, but maybe I will some day... Cyp 16:27, 18 Oct 2003 (UTC)
 * Support. Corrects my errors, adds good stuff :) Dysprosia 01:54, 19 Oct 2003 (UTC)
 * Make it so! --mav 22:46, 20 Oct 2003 (UTC)

You are now a Wikibooks admin. --mav

Hebrew Wikibooks
I want to promote the hebrew wikibooks, so i ask you to give me an admin premission in the hebrew wikibooks

--Mad dr 14:25, 14 Sep 2004 (UTC)

Please delete http://en.wikibooks.org/wiki/Orbital_Mechanics
http://en.wikibooks.org/wiki/Orbital_Mechanics and http://en.wikibooks.org/wiki/Physics/Orbital_Mechanics

I made a mistake in their creation. One of them should have been under Aeropace Engineering, and actually there is already and astrodynamics one which is the same.

SMW Guide
Hey, a lot of people are editing it wrong. Shouldn't you do something next time it happens? ( I took care of it.)--Nelson340 02:56, 19 Dec 2004 (UTC)

Programming:Windows Programming
This book is in pretty lousy condition, and I would like to start a project soonish (within the next few weeks) to try and expand this book. I have started a basic outline on my user page to try and lay out the things that (I think) this book should cover. I see that you have been a contributer to this book in the past, and I would therefore like to get your input on the current outline, and see if there is anything that i should add, or anything that I should remove from it.

The outline can be found at User:Whiteknight/Windows Programming. -- 23:37, 12 December 2005 (UTC)

De-Adminship vote.
A vote is being held to remove your sysop privileges. The basis for this vote is lack of use of these privileges. You may want to participate at Requests_for_adminship. See also the discussion at Staff_lounge. --JMRyan 23:13, 17 February 2006 (UTC)

Your De-Adminship
Hello. I would like to inform you that you will have your sysop rights removed here on 1 Nov 2006 due to inactivity. If you would like to discuss the matter, please see WB:RFA. You can re-apply for adminship at a later date if you wish. Thanks. -within focus 21:28, 1 October 2006 (UTC)

title: User talk:Cyp


text:
User:Cyp

en:User:Cyp

All my edits are also released under the proposed alternative licenses to the GNU, such as the creative something licence, or whatever they are all called, by the way, should this site decide to add an extra licence.

See also: User talk:Cyp

title: User:Cyp


text:
Wikibooks:Physics bookshelf

General

 * Modern Physics

Mechanics

 * Classical mechanics - This will be a "minimal standard course" of classical theoretical mechanics.
 * Mechanics

Energy and Power

 * Energy and Power - This course will be an introduction to energy and power within a societal context.

 * Nuclear Physics

Electromagnetism

 * Electrodynamics
 * Electromagnetism in the FHSST Physics book

Thermodynamics

 * Engineering Thermodynamics
 * Thermodynamics
 * Statistical Mechanics

Quantum mechanics

 * Quantum Mechanics
 * This Quantum World

Relativity

 * General Relativity
 * Special Relativity

Optics

 * Optics - Suitable for high school students.

Acoustics

 * Acoustics
 * Engineering Acoustics

Micro and Nanotechnology

 * Microtechnology - handbook of microfabrication and microtehcnology
 * Nanotechnology - handbook of nanoscience and nanotechnology
 * Carbon Nanotube Cookbook

Study Aids

 * A-level Physics
 * A-level Physics (Advancing Physics)
 * Physics Study Guide
 * Physics Exercises

History of Physics

 * Galileo's Science

Middle School

 * The Universe Made Simple

High School

 * [[File:100%.svg]] FHSST Physics - A free high school student textbook on physics
 * GCSE Science/Electricity - A simple book for students aged 14 - 16.
 * Optics - Suitable for high school students.
 * Physics with transforms - new invention - A new mnemonic for learning classical physics formulas using Euclidean geometry. It also serves to geometrically correlate classical physics formulas together. For high school & up.
 * High school physics - High school physics for the MTV literate.
 * SAT II Physics - Preparation for the College Board exam.
 * HSC Physics A physics textbook for Year 12 Australian high school students (Syllabus is closest to NSW)
 * Physics - For dummies, rebels and humans - A free textbook for Physics - the ultimate goal being to teach the student how to learn by himself.
 * Solved Question Papers - IIT JEE
 * Solved Question Papers - AIEEE

Hobbyist

 * Waves
 * LearnElectronics More hands on experience.
 * Table of nuclides - Table with all nuclides

Undergraduate

 * Physics Exercises
 * Electronics - Somewhat more advanced, for hobbyists or first or second year college level.
 * A-level Physics - For students taking Physics at A-level.
 * A-level Physics (Advancing Physics) - For students taking the OCR Physics B (Advancing Physics) course at A-level.
 * [[File:100%.svg]] Physics Study Guide - Overview of equations, basic terms, and variables for algebra-based physics course. - Suitable for A level/ advanced highschool or first year undergraduate.
 * Physics with Calculus - Undergraduate textbook, first- and second-year level
 * Physics practice questions (with solutions)

Advanced undergraduate

 * Mathematical methods of Physics
 * Modern Physics - A full-length undergraduate textbook. Includes Electromagnetism, Quantum Mechanics, and Thermodynamics and statistical mechanics.
 * Solid Mechanics
 * Introduction to Theoretical Physics - From First Principles to Newtonian Mechanics to General Relativity
 * Classical mechanics - This will be a "minimal standard course" of classical theoretical mechanics.
 * Engineering Thermodynamics
 * Heat Energy Transfer
 * Physics in the Language of Geometric Algebra. An Approach with the Algebra of Physical Space
 * Basic Physics of Nuclear Medicine
 * This Quantum World

Postgraduate

 * Interpretations of Quantum Mechanics and Quantum Field Theory
 * General relativity - Post graduate / advanced undergraduate
 * Quantum field theory - Postgraduate
 * Supersymmetry
 * String Theory & M-Theory

Related bookshelves

 * Miscellaneous natural sciences
 * Engineering
 * Mathematics

Wikinode: Other physics wiki

 * Qwiki is a quantum physics wiki devoted to the collective creation of content that is technical and useful to practicing scientists.
 * SklogWiki.org is a wiki for anyone interested in thermodynamics, statistical mechanics and the  computer simulation of materials, especially liquids and soft condensed matter.

Offsite physics books

 * Physics book series by Ben Crowell.
 * A Radically Modern Approach to Introductory Physics by David J. Raymond, GFDL license
 * Introduction to optics (PDF) Lecture notes from the University of Calgary. Includes some nice images.
 * The Slacker's Guide to Physics: Electricity and Magnetism
 * Structure and Interpretation Of Classical Mechanics A Lagrangian and Hamiltonian mechanics text using computational tools online
 * Lagrangian and Hamiltonian Mechanics: A short introduction Online at University of Arizona.
 * OPEN TEXT physics books
 * Introductory Physics by Charles Edward Pax, published under the GNU General Public License (GPL).
 * Grains of Mystique: Quantum Physics for the Layman by Thomas Haberkern and N Deepak, copyrighted 2002, licensed under Open Publication License http://www.ndeepak.info/qp
 * Lessons In Electric Circuits Copyright © 2000-2003, Tony R. Kuphaldt published under the "Design Science License", which allows for free copying, distribution, and/or modification of this document by the general public. http://www.ibiblio.org/kuphaldt/electricCircuits/
 * Calculus-Based Physics I & II by Jeffrey Schnick
 * Public domain, calculus-based physics book in PDF format.

de:Regal:Physik

title: Wikibooks:Physics bookshelf


text:
Waves/Transverse, Longitudinal and Torsional waves

Transverse and Longitudinal Waves

With the exception of light, waves are undulations in some material medium. For instance, waves on a slinky are either transverse, in that the motion of the material of the slinky is perpendicular to the orientation of the slinky, if you vibrate the slinky like a rope, or they are longitudinal, with material motion in the direction of the stretched slinky, if you treat it like a spring. (See image on right) Ocean waves are simultaneously transverse and longitudinal, the net effect being (nearly) circular undulations in the position of water parcels. The oscillations in neighboring parcels are phased such that a pattern moves across the ocean surface. Some media support only longitudinal waves, others support only transverse waves, while yet others support both types. Sound waves are purely longitudinal in gases and liquids, but can be either type in solids.

Mechanical transverse waves require a material medium and propogate by means of vibrations of the medium perpendicular to the direction of travel. Examples are water waves, ripples, seismic shear waves, and waves in stretched strings as above.

Electromagnetic (EM) waves (such as light) are also transverse waves but they do not require a medium and thus can pass through a vacuum (see intro). They consist of oscillating electric (E) and magnetic (B) fields which are perpendicular to the direction of propagation while also being mutually perpendicular. EM waves are a disturbance of space itself, which can be thought of as being stretched and therefore being elastic and having a tension. The B and E fields are in phase as shown on the left.

The fundamental S.I. unit of length (meter) is defined in terms of the speed of light in vacuum and the definition of the unit of time, the second. The previous definition was in terms of the wavelength of a particular color of light in the line spectrum of Krypton 86. The modern definition is more accurate as well as being the same to all observers regardless of their relative velocity.

Longitudinal waves propogate by means of vibrations or disturbances in the medium that are in the same direction that the wave travels. Examples are sound (as above), seismic shock waves, slinky springs and part of the motion in ocean waves. Sound waves are a series of high-pressure compressions and low-pressure rarefaction (for this reason, sound is sometimes called a pressure wave). While it might be convenient to think of individual molecules vibrating back and forth around an equilibrium position (producing areas of high and low pressure), the individual molecules in a gas generally move randomly, and it is only in large numbers that this pattern is visible. In longitudinal waves, the convention is to describe displacement in the direction the wave is going (the direction of propagation) as positive, and displacement against that direction as negative.

The pressure and displacement of a molecule at a point are $$90^o$$ out of phase, so that, for example, when a molecule is farthest displaced it is in a normal pressure area, while molecules in compressions or rarefactions have displacement close to zero.

Torsional waves consist of a twisting disturbance moving through a medium such as a wire or a rod.


text:
Waves/Sine Waves

Sine Waves
A particularly simple kind of wave, the sine wave, is illustrated in figure 1.2. This has the mathematical form:
 * $$h(x) = h_0 \sin ( 2 \pi x / \lambda ) ,$$ (2.1)

where
 * h is the displacement (which can be either longitudinal or transverse),
 * $$h_0$$ is the maximum displacement, sometimes called the amplitude of the wave,
 * λ is the wavelength.tory Physics fig 1.2.png|Figure 1.2: Sine wave]] -->

Figure 1.2: Definition sketch for a sine wave, showing the wavelength λ and the amplitude $$h_0$$ and the phase φ at various points.

So far we have only considered a sine wave as it appears at a particular time. All interesting waves move with time. The movement of a sine wave to the right, a distance $$d$$ may be accounted for by replacing $$x$$ in the above formula by $$x - d$$. If this movement occurs in time $$t$$, then the wave moves at velocity $$c = d/t$$. Solving this for $$d$$ and substituting yields a formula for the displacement of a sine wave as a function of both distance $$x$$ and time $$t$$:
 * $$ h(x) = h_0 \sin [ 2 \pi (x - ct) / \lambda ] . $$ (2.2)

The time for a wave to move one wavelength is called the period of the wave: $$T = \lambda / c$$. Thus, we can also write
 * $$h(x) = h_0 \sin [ 2 \pi ( x / \lambda - t / T ) ] . $$(2.3)

Physicists actually like to write the equation for a sine wave in a slightly simpler form. Defining the wavenumber as $$k = 2 \pi / \lambda$$ and the angular frequency as $$\omega = 2 \pi / T$$, we write
 * $$ h(x) = h_0 \sin (kx - \omega t) . $$(2.4)

We normally think of the frequency of oscillatory motion as the number of cycles completed per second and is given by $$f = 1/T$$. It is related to the angular frequency omega by $$\omega = 2 \pi f$$. The angular frequency is used because it is directly analogous to the wavenumber, see above. Converting between the two is not difficult. Frequency is measured in units of hertz, abbreviated Hz; $$1 \mbox{Hz} = 1 \mbox{ s}^{-1}$$ and angular frequency $$\omega$$ is in units of radians per second. (Not converting between frequency and angular frequency is a common mistake in many calculations)

The argument of the sine function is by definition an angle. We refer to this angle as the phase of the wave, $$\phi = kx - \omega t$$. The difference in the phase of a wave at fixed time over a distance of one wavelength is $$2 \pi$$, as is the difference in phase at fixed position over a time interval of one wave period.

As previously noted, we call $$h_0$$, the maximum displacement of the wave, the amplitude. Often we are interested in the intensity of a wave, which is defined as the square of the amplitude, $$I = h_0^2$$.

The wave speed we have defined above, $$c = \lambda /T$$, is actually called the phase speed. Since $$\lambda = 2 \pi /k$$ and $$T = 2 \pi / \omega$$, we can write the phase speed in terms of the angular frequency and the wavenumber:
 * $$c = \frac{\omega}{k} \mbox{(phase speed)} . $$ (2.5)

title: Waves/Sine Waves


text:
Waves/Types of Waves

Types of Waves
In order to make the above material more concrete, we now examine the characteristics of various types of waves which may be observed in the real world.

Ocean Surface Waves
Figure 1.3: Wave on an ocean of depth $$H$$. The wave is moving to the right and the particles of water at the surface move up and down as shown by the small vertical arrows. The particles move up, and gravity is the 'restoring' force. Water waves are also longitudinal, the water particles moving forward, then back. The restoring forces are more complex, but involve the inertia of the mass of water surrounding. Think of water waves as superimposed transverse and longitudinal waves. The net particle paths are nearly circular. This is typical of waves that travel along the boundary (interface) between two substances, in this case water and air.

These waves are manifested as undulations of the ocean surface as seen in figure 1.3. The speed of ocean waves is given by the formula
 * $$v = \left( \frac{g \tanh (kH)}{k} \right)^{1/2}, $$ (2.6)

where $$g = 9.8 \mbox{ m} \mbox{ s}^{-2}$$ is a constant related to the strength of the Earth's gravity, $$H$$ is the depth of the ocean, and the hyperbolic tangent is defined as 2.7
 * $$\tanh (x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} . $$ (2.7)

Figure 1.4: Plot of the function $$\tanh (x)$$. The dashed line shows our approximation $$\tanh (x) \approx x$$ for $$\vert x\vert << 1$$.

As figure 1.4 shows, for very small x, we can approximate the hyperbolic tangent by $$\tanh (x) \approx x$$, while for very large x it is positive 1 for positive x and negative 1 for negative x. This leads to two limits: Since $$x = k H$$, the shallow water limit, which occurs when $$kH << 1$$, yields a wave speed of
 * $$v \approx (gH)^{1/2}, \mbox{(shallow water waves)} , $$ (2.8)

while the deep water limit, which occurs when $$kH >> 1$$, yields
 * $$v \approx (g/k)^{1/2}, \mbox{(deep water waves)} . $$ (2.9)

Notice that the speed of shallow water waves depends only on the depth of the water and on $$g$$. In other words, all shallow water waves move at the same speed.

On the other hand, deep water waves of longer wavelength (and hence smaller wavenumber) move more rapidly than those with shorter wavelength. Waves for which the wave speed varies with wavelength are called dispersive. Thus, deep water waves are dispersive, while shallow water waves are non-dispersive.

For water waves with wavelengths of a few centimeters or less, surface tension becomes important to the dynamics of the waves. In the deep water case the wave speed at short wavelengths is actually given by the formula
 * $$ c = (g/k + A k )^{1/2} $$ (2.10)

where the constant $$A$$ is related to surface tension and depends on the surfaces involved. For an air-water interface near room temperature, $$A \approx 74 \mbox{ cm}^3 \mbox{ s}^{-2}$$.

title: Waves/Types of Waves


text:
Waves/Sound Waves

Sound Waves
Sound is a longitudinal compression-expansion wave through matter. The wave speed is
 * $$c = ( \gamma R T_{abs} )^{1/2} $$ (2.11)

where $$\gamma$$ and $$R$$ are constants and $$T_{abs}$$ is the absolute temperature. The absolute temperature is measured in Kelvins and is numerically given by
 * $$T_{abs} = T_C + 273.16^{\circ} $$ (2.12)

where $$T_C$$ is the temperature in Celsius degrees. The angular frequency of sound waves is thus given by
 * $$ \omega = ck = ( \gamma R T_{abs} )^{1/2} k . $$ (2.13)

The speed of sound in air at normal temperatures is about $$340 \mbox{ m} \mbox{ s}^{-1}$$.

title: Waves/Sound Waves


text:
Waves/Light

Light
Light moves in a vacuum at a speed of $$c_{vac} = 3 \times 10^8 \mbox{ m} \mbox{ s}^{-1}$$. In transparent materials it moves at a speed less than $$c_{vac}$$ by a factor $$n$$ which is called the refractive index of the material:
 * $$c = c_{vac} / n . $$

Often the refractive index takes the form
 * $$n^2 \approx 1 + \frac{A}{1 - (k/k_R )^2} ,$$

where $$k$$ is the wavenumber and $$k_R$$ and $$A$$ are constants characteristic of the material. The angular frequency of light in a transparent medium is thus
 * $$\omega = kc = \frac{k c_{vac}}{n} \approx

\frac{k c_{vac}}{\sqrt{1+A}}(1+ \frac{1}{2}\frac{A}{1+A}\frac{k^2}{k^2_R})$$

so the frequency increases slightly with increasing k. Typically, when k is near kR, the material becomes opaque.

Ultimately, this is due to resonance between the light and the atoms of the materials.

title: Waves/Light