Test 3

Bring

  • calculator
  • one 8 1/2 X 11 page of notes that you prepare ahead of time.

Topics

Double Integrals

Be able to

  • set up a double integral over an arbitrary area
  • do an integration in terms of the volume of a solid above an arbitrary area.
  • compute double integrals.
  • to determine limits of integration and change the order of integration.
  • Know what dA means and what dA is in rectangular and in polar coordinates.

Triple Integrals

Be able to

  • compute triple integrals.
  • set up triple integrals to compute the volume of a solid.
  • set up and compute triple integrals of a function of 3 variables over a solid region (Cartesian, cylindrical, and spherical).
  • change the order of integration and modify the limits appropriately.
  • Know what dV means and what dV is in rectangular, cylindrical, and spherical coordinates.

Coordinate transformations

Be able to

  • verify and use a coordinate transformation: $x(u,v,w), y(u,v,w), z(u,v,w)$; Check that it maps $(x,y,z)\mapsto (u,v,w)$ correctly.
  • calculate the Jacobian for a coordinate transformation, given the coordinate mapping, and know the interpretation of the Jacobian as the ratio of areas in the two spaces.

Vector Fields

Be able to

  • identify and draw vector fields.
  • test if a vector field is conservative, and, if so...
  • to find a potential function associated with that vector field.
  • Understand the relationship between gradient fields and level curves.
  • Know other consequences if a vector field to be conservative. (E.g. irrotational, path independence...)

Curl and Divergence

Be able to

  • compute the curl and divergence of a vector field.
  • do a graphical/visual analysis of a vector field, to find the sign of curl and divergence.
  • Understand what it means for a vector field to be irrotational or incompressible and be able to identify whether or not a vector field is irrotational or incompressible.

    Line Integrals

    Be able to

  • compute the line integral of a scalar function over a curve, including parameterizing the curve if necessary.
  • compute the line integral of a vector function (either the tangential component or the normal component) over a curve.
  • interpret "work that a field does on a particle" as the tangential component of a force field, $\myv F\cdot\uv T$, integrated over the trajectory of a particle.
  • estimate the sign of a vector line integral, based on the trajectory and a sketch of nearby vector field arrows.
  • Understand when a line integral is path independent and the relationships between path independence, gradient fields, conservative vector fields.

Green's Theorem

Be able to

  • Know the statement of Green's Theorem (both forms: in terms of tangential and normal components of vector fields).
  • apply Green's Theorem to convert line integrals $\leftrightarrow$ area integrals.
  • to use the divergence theorem to convert a surface flux integral (3-d) to a volume integral of divergence.