Day 33 in Mat329

• Your quiz is returned.
  • The first question just seeks to establish that you're ready to do integration in any of the three 3-dimensional coordinate systems.

    One of the most important question is units: it's a volume element. The product should have units of volume. Some of yours didn't.

    Secondly, I don't need to see any integrals. We're just talking about the volume element. You can put it in context, if you wish -- but this is about differentials, not integrals.

  • The second question was the doozy: if I asked you for the equation of a line, would you start integrating? Why do you try integrating when I ask for the equation of a cone?

    At the beginning of the class last time, I began by saying that the spherical coordinate system is particularly well-suited for describing spheres, and cones. But I'm like that teacher in the Charlie Brown specials: wah wah, wah wah, wah wah....:)

    Since so many of you wanted to compute a volume of a cone, let's do it.

• We have an exam Thursday, covering integration (and a little of the tail end of 14). So the sections covered include 14.7-8, 15.1-4. So six sections. Expect relatively uniform coverage.

• Your 15.8-9 homework is due today. Remember, we won't cover these two sections on the test.

It's a good time to try Jared's code again: Classify3D.nb -- let's hope that this one works! It's about maxes and mins....

• Last time I began by talking about multivariate functions with sodas as their function values. You might not have believed that such functions make sense. How about a function of two variables with political candidates as function values?.

• We now consider vector-valued functions in multivariate space. We've seen these before, however: if we have a multivariate function $f$ which is differentiable, then it has a gradient at every point, pointing to higher values of $f$.

  This is an important example: a gradient field.

• Not all vector fields are gradient fields, however: in general, we might think of a vector field F as ${\bf{F}}=\langle P(x,y),Q(x,y) \rangle$.

  Q: Is the vector field of Example 1 a gradient field?

  If F is a gradient field, then it is called a conservative vector field (and $f$ its potential function).

• Examples:
  • One more time: I altered Jared's code a bit, to provide insight into the gradient field, and its relationship to maxes and mins: Classify3DGradientField.nb

    This code plots the functions, with maxes and mins, but adds in the contour lines, and also the gradient lines.

    Let's use this to take a look at Example 6, and some other examples.

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  • #35 -- I've been plotting streamlines, so let's talk about them!:)