Consider the following simple example. Let f be a function of two variables defined by:
Let us define a function F of two variables by
Then we can calculate
With the obvious intentions regarding the partial derivatives we can also calculate
It can also be checked that
These two results are indeed true for a general function and substitution . They are regarded as a generalisation of the chain rule for one variable. Again and are often confusingly written as and .
The rule is then
It is very important to remember what is meant by all these items.
Note the chain rule can be used both ways.
ie let , , define
But also defines (substituting for u and v in terms of x and y)
and, of course, we put F as f throughout.
These results are special cases of the so called general chain rule.
If we have a function where x are substitutions
Consider the curve in which the surface meets the plane a constant.
In this plane , and would be a formula for the gradient of the tangent to the curve.
If we differentiate with respect to x, treating the y as if it was a constant (some say holding y constant), the the derivative obtained is called the partial derivative of with respect to x and we write or or . Similarily we have .
With functions of more than two variables one differentiates with respect to one of the variables by holding all the other variables constant.
In the obvious way we can have higher order partial derivatives.
Similarly for we have:
For commonly encountered functions we have that
From here on we may assume that all the mixed derivatives are equal.
Note, the normal rules (sum, product, quotient, function of a function) of differentiation apply to partial differentiation.
We can say that as the , or .
If, intuitively speaking, by going close enough to (a,b) we can get f(a,b) as close as we like to L.
f(x,y) is said to be continuous at (a,b) if .
Just as with one variable limits from below and above may be different, with two variables we may get various limits coming in from different paths.
Consider coming in to (0,0) along the line y=mx (m fixed). Along this line we have:
Thus along y=mx,
If we come in along the line ,
For a function of two (real) variables each element of its domain is an ordered pair of real numbers (x,y) only, and each element of its range is a real number, z, say.
x and y are called independent variables, and z the dependent variable.
To specify such functions we must have both a rule, and a domain.
If no known domain is specified we again take it to be the largest subset of for which the rule makes sense.
A graph of may be drawn in the usual way. (ie. (x,y) in the horizontal plane called the xy plane, and z moving along a third, vertical axis.
For obvious reasons informative graphs are often quite difficult to draw. It’s often useful to draw profiles of f, that is the surface curves where their surface meets planes parallel to the coordinate axis.
Consider the intersection with x=constant, y=constant and z=constant.
Another useful diagram of f(x,y) is that provided by drawing the contour lines (or level curves). That is we sketch in the x,y plane the graph of y=f(x,y) for various values of z.
Look at again.
For a function we can not sketch profile, and the best we can do is sketch level surfaces.
for any possible integer m we care to choose.
Choose then .
But as .
So, as .
So it can be said that increases more rapidly than any power of x.
Also (for ).
Where as , and as as .
So it can be said that increases more slowly than any positive power of x.
Also (for ).