1. We considered differentiating and integrating functions of a single-variable. How? We began with the notion of a limit, and then considered the derivative. If we have a, e.g., polynomial
Again we stop and reflect: this treats as if it were constant. So the derivative formed by
are “incomplete” or partial. There is some subtlety here due to using multiple variables, and we have to discuss the problems of limits first.
2. Definition. The function is “Continuous” at if
(i) is defined and finite;
(ii) is defined;
(iii) is defined (and finite).
Note: this can be determined by picking any curve which satisfies
for some , then taking
The subtletly here lies with being arbitrary. If two different curves produce two different results, the limit does not exist. Lets consider some examples and non-examples.
Example 1 (Limit Exists). Find
Solution: for this, we can simply plug in the values
This is because the function is sufficiently nice.
Example 2 (Limit Doesn’t Exist). What is
Solution: Lets first approach it along the -axis, i.e. first setting . We find
Now lets approach it on the -axis, i.e. first setting . We see
Still, approaching along the curve we see
But we have a problem: this implies . This cannot be! So the limit cannot exist! Very sad.
3. Definition. Let be defined on a region in the -plane, and let be an inerior point of , we just don’t want a boundary point!
exists, then it is called the “Partial Derivative” of at with respect to . It is denoted
evaluated at . NB: the subscripts in the indicate what variable we are taking the partial derivative of, i.e., it’s shorthand for .
Under similar conditions,
is the partial derivative of with respect to at . We denote this by
among a myriad of different conventions.
4. Higher order partial derivatives are done by taking it one at a time. So if
for example, we have
and taking its derivative again yields
Note we factor out in front of the partial derivative with respect to because is constant with respect to . So we then obtain
We take partial derivatives one at a time, from right to left:
Question: do partial derivatives commute? I.e., is always? Lets first consider an example calculation before considering an answer.
Example 3. Consider the function . Find .
Solution: We find that
Similarly, we find
Thus we conclude
5. Do Partial Derivatives Commute? Answer: not always. The conditions are fairly weak: if , , and are continuous throughout their respective domains, then
Exercise 1. Let where is some constant. Prove
Exercise 2. Let . What is ?
Exercise 3. Consider . What is ? What is ? Is ?
Exercise 4. Let . What is ? What is ?
Exercise 5. Let . What is ? What is ?