Title: Mathematics Review
1Mathematics Review
- Functions
- Derivatives
- Partial Derivatives
- Extremum (max and mins)
- Integration and Paths
2Functions 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)
3The 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?
4Comparing 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!
5Data 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
6How 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
7To 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
8The 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)
9Operations 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
10Multiple 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
11Partials 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
12Partials 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
13Comparing 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
14Some 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
15Extremum
- 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
16Extreme 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!
17Global 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
18Max, 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
?
?
?
19Rate 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
20Going 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?????
21Going 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
22Integration 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.
23Integration 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
24Integration 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