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.

- 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.

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)>:

- Compute limits, determine continuity, differentiate, and integrate.
- Compute velocity, acceleration, and speed.
- Compute unit tangent, unit normal, and the tangential and normal components of acceleration.
- Define arc length s(t).
- Explain the meaning of first and second derivative, unit tangent, unit normal, and curvature.

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

- Compute unit tangent, unit normal, and curvature.
- Explain the meaning of first and second derivative, unit tangent, unit normal, and curvature.

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:

- State the definition of continuity (three parts). Be able to classify discontinuities as removable and nonremovable. Be able to apply continuity properties.

- For partial derivatives (including higher derivatives): Define, interpret geometrically (first order) and as a rate of change, compute, know the different notations. Know the criterion for equality of mixed partials.

- Compute the differential, state the definition of differentiable, know conditions that imply (or do not imply) differentiable, "differentiable implies continuous", be able to use differentials for approximations.

- Be able to use the chain rule and do implicit differentiation.

For a function of two variables:

- Be able to define, compute, and explain the meaning of directional derivatives and gradient. Know the properties of the gradient.

- Be able to define tangent plane and normal line to a surface and determine their equations.

- Explain the meaning of relative and absolute extrema, state the extreme value theorem, know how to find critical points, be able to apply the second partials test and solve max-min problems.

- Constrained optimization: be able to use the Lagrange multiplier method.

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.

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.