- Return of the exam(s)
- Homework component
- You cannot show that a limit at a point exists by showing that the limit is the same from two directions. The limits under every approach must be equal!
- In-class component
- Take-home component
- the function
and the stereographic projection
- the function
- Homework component
- If you'll come and see me to discuss your test, you'll get 30% of your missed points back.
- Today's activity, and new assignment:
Fri 2/19 Discuss test; 20 - Read 21-23
- Homework to be determined.
- the function
- Last time you saw an example of a strange example: a function in some sense linear in the components of
(that is,
) which was nowhere differentiable wrt
- Example 2, p. 55 is even more bizarre (differentiable at a single point!)
- What conditions must
satisfy to be differentiable?
- As we've seen before, the rules for differentiation of functions of a complex variable parallel those of real variable functions,
- such as
- power rules
- constant multiples
- product rule
- quotient rule
- chain rule
- The derivations flow right from calculus (e.g. the product rule)
- Your most recent homework includes problems relevant to this section. Come with questions, if necessary.
- Well, we can get necessary conditions, by considering the limit definition of the derivative and then two different approaches: in order for the derivative to exist, we must get the same values along the two different approaches.
- In the end, we find that the following is necessary for a
function to be differentiable at a point: the partial derivatives of
the component functions must satisfy
- Check:
- A polynomial (e.g.
- That troublesome function
- A polynomial (e.g.