|Welcome to Math 257
||Notes and Links
||About This Class; Day 1 Handout (pdf, html); Monday, Wednesday, Friday, Day 2 Handout (pdf, html); First Week Notes.
||Monday, Wednesday, Tutorial 2 Handout, Friday, Second week notes, HW1, HW1 Solutions.
||Monday, Wednesday, Tutorial 3 Handout, Friday, Third week notes, Class Photo, HW2, HW2 Solutions.
||Monday, Wednesday, Friday, 4th week notes, HW3, HW3 Solutions.
||Monday is Thanksgiving, no class; Wednesday, Friday, 5th week notes, HW4, HW4 Solutions.
||Monday, Wednesday, Friday, 6th week notes, HW5, HW5 Solutions.
||Monday, Wednesday, Friday, 7th week notes
||Monday; Term test 1; Wednesday, HW6, HW6 pdf, HW6 Solutions, Friday, 8th week notes.
||Monday is last day to switch to MAT 237; Monday-Tuesday is UofT Fall Break; Wednesday, HW7, HW7 Solutions, Friday, 9th week notes.
||Monday, Wednesday, HW8, HW8 pdf, HW8 Solutions, Friday, Lecture recordings, 10th week notes.
||Monday, Wednesday, HW9, HW9 pdf, HW9 Solutions, Friday, Lecture recordings, 11th week notes.
||Monday, Wednesday, HW10, HW10 Solutions,makeup class on Thursday at GB 120 at 5PM, no class and no DBN office hours Friday! 12th week notes.
||Monday, Wednesday, 13th week notes Semester ends on Wednesday - no class Friday.
||No classes: other classes' finals, winter break.
||Class resumes Friday at RS211, no tutorials or office hours this week, Friday, Friday notes.
||Monday, Wednesday, Friday, Weekly notes
||Monday, Term test 2; Wednesday, HW11, HW11 inline pdf,HW11 solutions, Friday, Weekly notes
||Monday; Hour 44 Handout (pdf, html); Wednesday, HW12, HW12 inline pdf, HW12 Solutions, Friday, Weekly notes
||Monday, Wednesday, HW13, HW13 inline pdf, HW13 Solutions,Friday, Weekly Notes
||Monday, Wednesday, HW14, HW14 inline pdf, Friday, Weekly Notes
||Monday, Wednesday, HW15, HW15 inline pdf, HW15 solutions, Friday,Weekly Notes, UofT examination table posted on Friday.
||Reading week - no classes; Tuesday is the last day to drop this class.
||Monday, Wednesday, HW16, HW16 inline pdf, Friday, Weekly Notes
||Monday, Wednesday, Friday, Weekly Notes
||Monday, Term test 3 on Tuesday at 5-7PM; Wednesday, HW17, HW17 inline pdf, Friday, Weekly Notes
||Monday, Wednesday, HW18, Friday
||No Monday class, Wednesday, Thursday (makeup for Monday, 5PM at MP 134), HW19, Friday
||Monday, Wednesday; Semester ends on Wednesday - no tutorials Wednesday and Thursday, no class Friday.
||The Final Exam.
|Register of Good Deeds
Add your name / see who's in!
||Dror's notes above / Students' notes below
Symmetry of Second Partial Derivatives
(Note: This is based off of the proof in the textbook, and may be slightly different from how it was presented in lecture.)
Let be an open subset of , and let be of class . Then, for all , we have that .
We begin by defining a function that will help us in our proof. Let be an arbitrary point. We then define the function as follows:
Why do we define such a function? In fact, we can define both of the partial derivatives in question in terms of :
And similarly for , but with the two limits taken in the opposite order.
Now, let's make use of this function. Let us consider the square , where and are so small that . We will show that there exist points such that:
The proofs of the two equalities are symmetric, and thus we only explicitly prove the first one. We will prove this through a double application of the mean value theorem.
Consider the function , defined such that . Then . By hypothesis, exists at all points of , so we can differentiate with respect to the first variable of on the interval . By the mean value theorem, this means that there exists a point such that .
Now we want to apply the mean value theorem one more time. Consider the function , defined such that . By hypothesis, exists at all points of , so we can differentiate with respect to the second variable of on the interval . So, by the mean value theorem, there exists a point such that . Thus:
Now we can prove the theorem. Let , and let be so small that . By what we have just shown, , for some .
is the length of the sides of the rectangle , so as , . By hypothesis, is continuous, and so as , . Thus:
We can use the same argument, and the second equality from step 2, to show that:
Therefore, by the uniqueness of limits, the two quantities must be equal.
Handwritten Lecture Notes in PDF
MAT257 - Lecture14 (Oct 14)