# 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.