Day 32 in Mat329

• We do have an exam next week, Thursday, covering integration (and a little of the tail end of 14).
• Your 15.8-9 homework is due Tuesday.

Yesterday we finished cylindrical coordinates. In every event, you need to think about how best to represent the situation that you're dealing with, and symmetry and structure may indicate that a particular coordinate system is particularly sensible.

Problems always start out the same, e.g. $W=\int\!\int\!\int\!dW$ -- then our job is to figure out how best to rig things so that the limits of integration, and the differential $dW$ make doing the integral as easy as possible.

We'll just look at some problems that have spherical symmetry that we might exploit. Figures 5 and 8 are worth examining in some detail.... \[ x=\rho\sin{\phi}\cos{\theta} \hspace{1in} y=\rho\sin{\phi}\sin{\theta} \hspace{1in} z=\rho\cos{\phi} \] and the volume element becomes \[ dV = \rho^2\ \sin(\phi)\ d\rho\ d\theta\ d\phi \]

• #1a
• #3b
• #8
• #21
• #38

• 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:
  • #1, 1085
  • #2
  • #11-14