SUNY Geneseo Department of Mathematics
Monday, October 31
Math 223 01
Fall 2022
Prof. Doug Baldwin
The heated sphere problem (problem 2) in problem set 8.
The first part of the question mostly requires figuring out what the description of the temperature function means in mathematical terms. Key parts are “distance from origin” (use the distance formula), “inversely” (the distance will appear in the denominator of a fraction), and “proportional” (the actual function involves a constant of proportionality that we don’t know yet):
Once you have the basic form of the temperature function, you can plug in point (1, 2, 2) for (x, y, z) and solve for the constant of proportionality.
In the second part of the problem, you need to find the directional derivative of the temperature function in the direction between the two given points:
With Prof. Kochalski leaving the math department, I’m taking over her calculus course.
There should still be enough time on my schedule for all of you to make the appointments you need, but there will be less of it. You can’t count as much on being able to make appointments at the last minute, and it will be especially dangerous to try to grade a lot of problem sets in the last few weeks of the semester — stay caught up!
Being as this is Halloween week, I have a bowl of candy in my office. Anyone who comes to my office to grade something this week can have some.
Problem set 9, on optimization of multivariable functions, is ready.
Work on it this week and grade it next week.
See the handout for more information.
One question asks you to use Mathematica to help solve an absolute extreme value problem. Two features we haven’t yet talked about may be helpful for this: user-defined functions, and the built-in Solve function.
You can download a notebook in which I demonstrated the basics of both features.
Based on “Volumes and Double Integrals,” “Properties of Double Integrals,” and “Iterated Integrals” in section 4.1 of the textbook.
Definition: a double (or more dimensional) integral is the limit of a double (or more dimensional) Riemann sum.
Evaluate multiple integrals as “iterated integrals” via Fubini’s Theorem. Specifically, integrate first with respect to 1 variable, then integrate result with respect the other.
Most properties of single-variable integrals apply to multivariable integrals.
A 2-variable definite integral is defined as a limit of a double sum:
How do you suppose this extends to 3-variable, 4-variable, etc. functions?
Just add an additional summation for each dimension:
Integrate xy2 over the region 0 ≤ x ≤ 2, 0 ≤ y ≤ 1.
First we decided on an order of integration. We decided to integrate with respect to x first, then y. Then we integrated, working from the innermost integral out. In general, each integral evaluates to an expression involving the outer variables, which then gets integrated in turn.
Notice that you could also have evaluated this example as a product of 2 single-variable integrals, thanks to one of the properties given in the book.
Some applications of multivariable integrals.
Please read “Applications of Double Integrals” in section 4.1 of the textbook.