Performance Expectations
Calculus 3

Vectors and Geometry of Space | Vector-valued Functions | Functions of Several Variables | Multiple Integrals | Vector Analysis

The Mathematics Department has a standard of performance expected of calculus students. The following pages describe in detail what you need to be able to do and what you need to understand to receive an A in Calculus 2. This provides an excellent checklist you can use each day to make sure you have learned the material well enough. Before an exam you can use the list of expectations as a checklist to make sure you have reviewed everything.

Vectors and Geometry of Space

Definitions: Be able to state precisely the definitions of each of the following in 3-dimensional euclidean space.
vector addition
scalar multiplication of a vector
length of a vector
basis unit vectors i, j ,k
parallel vectors
orthogonal vectors
cylinder (cylindrical surface)
trace of a surface in the xy, xz, and yz-coordinate planes
Know how to use
properties of dot product
algebraic and geometric properties of cross product and triple scalar product
cylindrical and spherical coordinates for plotting points and graphing a function
Know how to do
Vectors: perform vector addition and scalar multiplication, compute the length of a vector and the length of a scalar multiple of a vector, find a unit vector in the same direction as a given vector, find the component form of a vector with specified length and direcion, compute the distance between two points in 3-space, find the component form of a vector from one point to another point in 3-space.
Dot product: compute the angle between two vectors, determine whether two vectors are orthogonal, compute the scalar projection of a vector in the direction of specified vector and the (vector) component of a vector in the direction of a specified vector.
Cross product: compute cross product using a 3 by 3 determinant, compute triple scalar product using a 3 by 3 determinant, draw the vector diagram illustrating a cross product or triple scalar product .
Lines in space: express the parametric and symmetric equations for a line in space and know what starting information is necessary to write these equations, compute the cosine of the angle between two lines, compute the distance between a point and a line.
Planes: express the standard and general equations for a plane and know what starting information is necessary to write these equations, compute the angle between two planes, compute the distance between a point and a plane.
Surfaces in space: describe the graph of a cylinder from its equation, identify one of the six quadric surfaces from its equation, specify an equation for a surface of revolution given an equation for its generating curve.
Cylindrical and spherical coordinates: graph a surface given its equation, convert between cylindrical or spherical coordinates and rectangular coordinates.

Vector-valued Functions

For a space curve r(t), identify and describe its graph (e.g. helix).

For a vector-valued function r(t) = <x(t), y(t), z(t)>:

For a vector-valued function r(s) = <x(s), y(s), z(s)>:

Functions of Several Variables

For a function of several variables, know the meaning of domain, range, level curves (surfaces), traces and how to find them.

In the plane know the meaning of neighborhood, interior point, open set, boundary point, closed set. Be able to compute limits of functions.

For a function of several variables:

For a function of two variables:

Multiple Integrals

Be able to set up limits of integration for a plane region and to evaluate the associated iterated integral.

Know how to compute the area of a plane region using a double integral.

Know and be able to use the properties of double integrals.

Be able to change the order of integration for a double integral.

Know how to set up and evaluate a double integral in polar coordinates.

Be able to convert a double integral from rectangular to polar coordinates.

For a planar lamina, be able to do computations involving mass, moments, angular speed, kinetic energy, moment of inertia, and radius of gyration.

Be able to set up and evaluate integrals for surface area.

For triple integrals, know how to evaluate using iterated integrals, be able to compute volume of a solid, be able to change order of integration.

Know the volume elements dV for cylindrical and spherical coordinates, be able to set up and evaluate triple integrals with these coordinates.

Know how to compute a Jacobian and how to use it for a change of variables in a double integral.

Vector Analysis

For a vector field: be able to test whether it is conservative, find its potential function, and compute divergence and curl.

Line integrals: Know the definitions of smooth curve and piece-wise-smooth curve, be able to parametrize curves, evaluate line integrals, and evaluate work integrals. Know the various forms of the integrand for a work integral.

Conservative vector fields: be able to apply the fundamental theorem of line integrals, know the equivalent conditions for independence of path, know the law of the conservation of energy.

Green's theorem: Be able to identify a simply connected region, be able to apply Green's theorem and the extended version of Green's theorem.

Surface integrals: know how to parametrize surfaces, compute unit normals, identify flux-type integrals and evaluate them.

Divergence theorem: Be able to apply Gauss' divergence theorem.

Stoke's theorem: Be able to apply Stoke's theorem.