Mathematics Review

1
Mathematics Review

• Functions
• Derivatives
• Partial Derivatives
• Extremum (max and mins)
• Integration and Paths

2
Functions of Variables

• A goal of science is to explain the observed via
  theories
• Allows you to predict the unobserved with some
  certainty
• The observed are usually measured quantities
  like T or P
• These are physical variables
• In the end, our explanation usually takes the
  form of a relationship of the possible variables
  that describe the properties of what we study
• Empirical relationship found by examining
  experimental data
• Trends in data or graphs of data show
  relationships of variables
• Analytic relationship found via theoretical
  means
• Within reasonable error, experimental data is
  always correct, so theory must match it
• Relationships between the variables takes the
  form of a function of variables, usually called
  an equation of state
• For example, y f(x) for y ln(x)
• Specifically, PV nRT says that P f(n,T,V)

3
The Function

• Mathematically, a function is a way to map one
  value to another through a predefined
  algorithm/relationship
• If y 2x and x 2, then the value of y is 4.

y-domain
x-domain

• Well use a slightly less rigorous concept of a
  function
• For us, a function will hold the relationships
  between the real, physical properties of whatever
  we are studying
• With luck, this relationship will hold for many
  different types of stuff that we will study
• For example, KE ½mv2 says, if you give me the
  velocity mass of any object, then I can tell
  you its kinetic energy
• This relationship is linear with respect to mass
• But its nonlinear with respect to velocity
• What about with respect to v2?

4
Comparing Theory and Experiment

• Once we have data from experiment, we will
  usually compare to theory via a graph
• Compare overall trends
• Compare physically important points

Good fit between the experiment and
theory. Linear relationship between y and x!
Poor fit between the experiment and
theory. Linear theory with nonlinear
experimental evidence!
In these examples, theory is shown as the line
while experiment is shown as data points.
Good fit between the experiment and
theory. Nonlinear relationship between y and x!
5
Data and Graphing

• Understanding graphs is paramount
• What does a graph tell you?
• The x-axis data is what you control
• Called the independent variable
• You set this and then measure y
• The y-axis data is what changes in response to
  the conditions you set
• Called the dependent variable

• Even nonlinear data can be made to look linear
• Linear trends are consistent
• This requires a change in axis
• For the above example y ex, so change x-axis to
  ex and will get a linear relationship

6
How a Function Changes

• How do we describe the change of a function?
• Whether linear or nonlinear!
• Lets measure the change as a ratio of the change
  in the y direction as compared to the change in
  the x direction
• To do the math, simply take Dy/Dx, or take the
  slope!
• This is easy for a linear relationship, but what
  if nonlinear?
• Choose a point on the curve and draw a tangent
  line to it
• The slope of the tangent line is how the function
  changes here
• This value changes depending on where we choose
  to go tangent
• This is a concrete way to identify a nonlinear
  function

Dx
y
y mx b
Dy
x
7
To Infinity and Beyond

• For our nonlinear example, it seems awfully
  tedious to find the rate of change of our data
• Newton recognized this and found a simpler method
• He said that if we squeeze the Dx and Dy down
  infinitely small (make the line smaller and
  smaller), then the resulting line will be linear
  on the actual scale of the graph

• If we had a way to get a function that did this
  same manipulation, then the function would be
  able to tell us this same story (the slope at any
  point)
• Newton found that the derivative operation
  created a function that did exactly this
  manipulation
• In short, this operation creates a function that
  shows us how the original function moves along
  its path

8
The Derivative

• Taking the derivative of a function
• Shows rate of change of f(x) w/respect to some
  variable
• Gives us either
• Slope of line tangent to some point on f(x)
• Inflection point of f(x)extremum
• There are quite a few flavors of derivatives out
  there

For example, y xln(wz)
9
Operations I The Derivative

• The derivative is an operator, just like , -, ,
  or .
• If y is a function of x, then we would write the
  derivative as

• Similarly, if Pf(V), as in the ideal gas law,
  then we could find how P changes with changing V!

• All we need to know is how to do this operation
  (d/dV)
• So, lets try a few and rekindle our derivative
  ability

10
Multiple Variables Add a Dimension

• So far, all our functions had only two variables
  (x and y, t and b, etc), but most physically
  meaningful parameters are related to more than
  just a single other parameter
• So consider some function of both x and y, or z
  f(x,y)
• How does the derivative work here?
• Lets start with a graphical exploration as we
  did before

z

• Obviously, this curve exists in three dimensions
• That is, a change in z is in response to changes
  in either x or y, or both x and y!
• Lets imagine a portion of the line moving in the
  yz-plane only
• x is constant under these conditions
• And z f(y) only, so could write

y
x
If large change
If small change
11
Partials I

• In our 3D example, we had to hold one variable
  constant to get the familiar y mxb formula,
  but we didnt tell people
• There must be a way to tell folks that the change
  were describing is ignoring one of the available
  dimensions
• So, we write the derivatives not as infinitesimal
  derivatives but as partial derivatives
• The real way to write down how z changes is thus

The total differential!

• This works for all possible cases. But why???
• Well, if by chance only one variable changes,
  then the other variables derivative is zero
• i.e. if y is constant, then dy 0
• The above expression is good for any number of
  variables and can be written immediately, without
  thought
• If G f(H,T,S), then

12
Partials II

• Lets break this total differential down, once
  again

• dG the change in G
• dH the change in H

• dT the change in T
• dS the change in S

13
Comparing Information

• OK, so we have this total differential for the
  variable G
• Works for any exact differential (i.e. state
  functions)
• Suppose we know how G is related to H, T, and S
• Let it be GH-TS
• Then take the derivative of this form and see
  what happens
• To do this, take the derivative in turn using the
  product rule
• That is, take the derivative with respect to H,
  then with respect to T, then with respect to S,
  and then add them up
• Now compare to what we had earlier

Derivative Total differential

• The above skills (in red) will be very useful all
  semester

14
Some Partial Examples

• Partial derivatives are just like any other
• Except there are other variables in the function
• You simply ignore the changes of the other
  variables
• Treat them like constants
• Some examples

15
Extremum

• Since the derivative operation yields a function
  that tells us how the parent function changes at
  any point, it might be able to tell us when the
  direction of change, changes
• When does the function stop increasing and begin
  decreasing
• In other words, what are the maximum and minimum
• Lets look at this graphically
• Imagine a portion of a function where its
  obvious a maximum occurs

• Leading to the maximum, all tangent lines have
  positive slope
• Leading away, all tangent lines have a negative
  slope
• But at the maximum, what is the slope?
• This line is horizontal, so dy 0
• This means that the slope 0
• Aha, the slope equals zero at a maximum
• But its also true for a minimum
• And a saddle point

16
Extreme Mathematics

• If y f(x), then the extremum can thus be found
  with

• You simply take the derivative, set it equal to
  zero and solve for x to find the point(s) that
  represent extremum
• For example, take y 3x2 5x

• The derivative is y 6x 5 0
• We thus know that x is an extremum at x -5/6

• Plotting the derivative itself, we see

that x does equal zero when y equals -5/6!
17
Global Considerations

• There is a problem with the derivative test.
  Some functions have many extremum
• Some max, some min, some saddle points
• And what if your extreme is in another dimension
• That is, what if you take d/dx and the real min
  is in d/dy?
• We better check in all dimensions for
  multidimensional data
• The picture were talking about is like this

• Here we see a local minimum, and
• A global minimum

Local min
Global min

• Much research goes into efficient methods to
  search through a function space to locate global
  extremum
• Max and mins are common areas of interest in
  chemistry, physics, and biology

18
Max, Min, or Saddle Point?

• So you can locate an extremum. It would be nice
  if you could also tell (without a picture) which
  type it is.
• Lets look again at a picture to aid us
• If the derivative tells us the change
• What if watched how the change, changes?
• There is an obvious maximum at 2, so lets look
  just before and just after for a clue
• Just before, the change is positive (tangent line
  with slope)
• Just after, the change is negative (tangent line
  with slope)
• The change of the change is D(change) changef
  changei
• D(change) - - () -
• So, we just determined that a maximum of a
  function must be a change from a positive slope
  to a negative slope
• But a change is just a derivative!!!
• Finding the max must be equivalent to finding out
  the sign of the second derivative of the
  function! If its negative, then the extremum is
  a maximum

?
?
?
19
Rate of Change of Slope

• By symmetry, we can deduce that a minimum must be
  present when the second derivative is positive!
• But what is meant by this? How do we
  mathematically find a value to use?
• For any function
• Take the derivative, set to zero and solve for
  changing variable
• Take the 2nd derivative plug above values in
  for the variable
• If this yields a positive number, then this point
  is a minimum
• If this yields a negative number, then its a
  maximum
• If this yields zero, then its a saddle point
• Heres an example A b3 - 12b2

20
Going Backwards A Question

• Imagine a ball moving through the air

• Its position at any time is x
• Derivative with time, v, tells us how its
  position is changing
• Derivative with time again, a, tells us how the
  velocity is changing
• Not all changes get special names, but some do
• If we have a function to represent the position
  at any time (its path), we could predict v and a
  at any future time

• Lets go the opposite direction, now
• Imagine we can measure the acceleration, a, at
  any time
• If we had this function, what could we do with
  it?
• Could we ever get the path just from this
  information?
• In other words, a function can be manipulated
  with the derivative to give us useful
  information. But what if our function represents
  a physical property that is already a derivative
  of another physical property?????

21
Going Backwards The Integral

• Heres another view of the physics from the
  previous page

Position (x) Velocity (v) Acceleration (a)


1st Derivative
Integral 2


integration
differentiation
2nd Derivative
Integral 1

• Going from top to bottom, we take the derivative
  (twice)
• Going the other direction, from bottom to top, is
  like undoing the derivative
• Its anti the derivative and undoes this
  operation
• So call it the antiderivative, or, better yet,
  the integral
• Well find this useful when we know how something
  changes (have a function for it) but want to know
  the actual value of the thing that is changing

22
Integration I

• Well do very few integrals in this semester, so
  lets just learn the very basics
• First, integrals are symbolized as
• The dx is an infinitesimal change in x
• The variable that were unchanging is x
• The f(x) is the functional form of the slope of
  some other functionthe one were after
• This notation comes from the derivative notation

Separating the variables of dy/dx
The derivative notation is

• Now, we integrate both sides (use the curly
  symbol above)

Note weve just shown the simplest integral.
The integral of a differential yields the
variable itself.
23
Integration II

• The difficulty lies in undoing the derivative of
  the functional part of the integral
• This f(x) can be a function or a constant, too!
• Here are some antiderivatives you should be aware
  of
• In general, they all follow this form
• Or this form

c constant, because
24
Integration III

• Any integration must occur over some range of the
  variable
• Try this real example
• A block with mass m has a velocity depending on x
  such that its velocity follows y kx3. What is
  the path of this block as it goes from A to B?
• First, imagine the a graph of the block moving
• Obviously, A represents some (x, y) pair
• And B also represents some (x, y) pair
• So, moving from A to B means x y change
• We denote each change in the limits of integration