Syllabus

express vectors with appropriate notation.

perform the following basic operations on vectors: add, subtract, scale, find the magnitude, and find the direction.

use vectors to solve application problems. (§ 13.1, 13.2)

compute the dot product of two vectors and use this to compute the angle between two vectors.

determine whether two vectors are orthogonal, compute the orthogonal projection of one vector onto another, and use these to solve application problems. (§ 13.3)

compute the cross product of two given vectors to find a third vector orthogonal to each of the given vectors.

use the cross product to solve application problems. (§ 13.4)

find the equation of a line in space given either a point and a vector in space, or two points in space.

express the line as a vector equation and using parametric equations.

solve application problems involving lines in space. (§ 13.5)

find the equation of a plane given either a point and a normal vector in space, or three points in space.

express this as a vector equation and as a linear equation in three variables. (§ 13.5)

match standard quadric surfaces and their standard equations.

place a given equation for a quadratic surface into standard form. (§ 13.6)

sketch graph of a vector-valued function of a single variable.

compute limits involving vector-valued functions and determine whether a vector-valued function is continuous at a point. (§ 14.1)

use the derivative rules for vector-valued functions to find the vector tangent to a point on a space curve.

compute the unit tangent vector at a point on a space curve.

compute definite and indefinite integrals involving vector-valued functions. (§ 14.2)

represent the path of an object moving in space using a vector-valued function and compute its velocity, speed, and acceleration.

find the vector-valued function representing the path of an object moving in space given either the acceleration or velocity vector and initial conditions.

recognize straight-line and circular motion, and solve application problems. (§ 14.3)

compute the length of a space curve on a closed interval.

parameterize a curve by arc length under appropriate conditions. (§ 14.4)

compute the curvature, principal unit normal vector for a smooth space curve, and use these to decompose the acceleration.

compute the binormal vector, interpret the \(\mathbf{TNB}\) frame, and compute the torsion for a smooth space curve. (§ 14.5)

find the domain and range of a function of two real variables.

sketch level curves and surfaces for functions of two real variables. (§ 15.1)

use the limit laws for functions of two real variables to compute limits and determine whether a function is continuous at a point in the domain.

I know the definition of interior point, boundary point, open set, and closed set. (§ 15.2)

compute the partial derivatives for functions of two or more variables.

determine whether a function of two variables is differentiable. (§ 15.3)

use the chain rule on functions with one or more independent variables.

use implicit differentiation on a function of two variables.

solve application problems involving these techniques. (§ 15.4)

compute the gradient and directional derivatives of a function of two or more variables.

use these techniques to solve application problems. (§ 15.5)

compute the plane tangent to a surface at a point from either the implicit or explicit form.

approximate a surface near a point using the tangent plane.

work with differentials and solve application problems with these techniques. (§ 15.6)

identify critical points of functions of two variables.

use the second derivative test to classify critical points and find maximum/minimum values on closed bounded sets. (§ 15.7)

use Lagrange Multipliers to find absolute extrema on closed and bounded constraint curves. (§ 15.8)

use iterated integrals to compute double integrals over a region in \(\mathbb{R}^2\)

identify when to use polar coordinates to simplify these iterated integrals. (§ 16.1, 16.2, 16.3)

use iterated integrals to compute triple integrals over a region in \(\mathbb{R}^3\)

identify when to use cylindrical and spherical coordinates to simplify these iterated integrals. (§ 16.4, 16.5)

compute the center of mass of an object (§ 16.6).

use a change of variables in two and three dimensions to simplify an interal (§ 16.7).

match vector fields with their graphs.

parameterize curves, evaluate line integrals, and interpret these as mass, work, flux, and circulation. (§ 17.1, 17.2)

determine whether a vector field is conservative and, if so, find the potential function.

use the Fundamental Theorem for Line Integrals to evaluate line integrals of conservative vector fields. (§ 17.3)

use Green’s Theorem to evaluate line integrals of vector fields. (§ 17.4)

compute the divergence and curl of a vector field. (§ 17.5)

parameterize a given surface and compute surface integrals. (§ 17.6)

use Stoke’s Theorem to evaluate integrals (§ 17.7).

use the Divergence Theorem to evaluate integrals. (§ 17.8).