ISU Mathematics Department

Objectives for Calculus III

Geometry in Space, Vectors

  • Use the parallelogram law to add geometric vectors.
    Resolve geometric vectors into components parallel to coordinate axes.
  • Perform the operations of vector addition and scalar multiplication, and interpret them geometrically.
  • Use the dot product to calculate magnitude of a vector, angle between vectors,
    and projection of one vector on another.
  • Find and use direction angles and direction cosines of a vector.
  • Use parametric equations for plane curves and space curves.
  • Use and convert between parametric and symmetric equations for a straight line.
  • Find a tangent line at a point on a parametric curve; compute the length of a parametric curve.
  • Compute velocity, unit tangent and acceleration vectors along a parametric curve;
    resolve acceleration into tangential and normal components and compute curvature.
  • Use and interpret geometrically the standard equation for a plane.
  • Use the cross product; interpret the cross product geometrically and as area of a parallelogram;
    interpret the vector triple product as volume of a parallelopiped.
  • Recognize cylinders and quadric surfaces from their Cartesian equations.
  • Use cylindrical and spherical coordinates, and convert among these two and rectangular coordinates.

Derivatives for Functions of Two or More Variables

  • Represent a function of two variables as the graph of a surface; sketch level curves.
  • Calculate partial derivatives and the gradient.
  • Use the gradient to find tangent planes, directional derivatives and linear approximations.
    Interpret the gradient geometrically.
  • Use the Chain Rule.
  • Find and classify critical points of functions, using the second derivative test.
  • Find maximum and minimum values for a function defined on a closed, bounded, planar set.

Multiple Integrals

  • State the definition of the integral of a function over a rectangle.
  • Use iterated integrals to evaluate integrals over planar regions, and to calculate volume.
  • Build on elementary integration techniques to evaluate multiple integrals efficiently.
  • Set up and evaluate double integrals in polar coordinates.
  • Set up and evaluate integrals to compute surface area.
  • Set up and evaluate triple integrals in Cartesian coordinates.
  • Use double and triple integrals to compute moments, center of mass, and moments of inertia.
  • Use cylindrical and spherical coordinates; change coordinates from
    rectangular to cylindrical or spherical or the reverse.
  • Set up and evaluate triple integrals in cylindrical and spherical coordinates.
  • Change the order of variables in multiple integrals.
  • Carry out change of variables in multiple integrals.

Vector Calculus

  • Calculate the curl and divergence of a vector field.
  • Set up and evaluate line integrals of scalar functions or vector fields along curves.
  • Recognize conservative vector fields, and apply the fundamental
    theorem for line integrals of conservative vector fields.
  • State and apply Green's Theorem.
  • Set up and evaluate surface integrals; compute surface area and
    the flux of a vector field through a surface.
  • Set up and evaluate integrals over parametric surfaces.
  • State and apply the Divergence Theorem.
  • State and apply Stokes' Theorem.