Math 280 Course Goals and Objectives

In successfully completing this course, a well-prepared student should

• understand differential and integral caculus of vector-valued functions F:R→Rⁿ
• understand differential and integral caculus of functions of several variables f:Rⁿ→R
• understand differential and integral caculus of vector fields F:Rⁿ→Rⁿ

In terms of specific objectives, a well-prepared student should be able to

• state a definition for each relevant term (e.g., cross product, partial derivative, triple integral)
• understand and draw relevant pictures for geometric definitions of vector, magnitude, addition of vectors, multiplication of a vector by a scalar
• compute with component expressions for given vectors
• give a geometric definition of dot product
• compute a dot product using component expressions for two given vectors
• use the dot product to compute the angle between two given vectors
• compute the component or projection of a given vector in the direction of a second given vector
• give a geometric definition of cross product
• compute a cross product using component expressions for two given vectors
• use the cross product to find a vector perpendicular to two given vectors or to compute the area of a given parallelogram
• use algebraic properties and identities for vector addition, scalar multiplication, dot product, and cross product to simplify or rewrite a given vector expression
• prove a given vector identity
• understand the connections among the various forms for the equation of a plane, including the point-normal form
• find the equation of a plane using given information (for example, the coordinates of three points)
• find the parametric equation or symmetric equations of a line given information (for example, the coordinates of two points on the line)
• compute the distance between two given geometric objects (for example, between a point and a plane)
• use coordinate equations for lines and planes to analyze geometric questions (e.g., finding the point at which a given line intersects a given plane)
• use traces and cross-sections to determine the surface described by an implicit equation in three variables
• determine the domain of a vector-output function and plot or describe the output curve for a simple vector-output function
• give a linear vector-output function to parametrize a given line
• parametrize a simple curve described geometrically
• analyze a limit of a vector-output function
• analyze continuity of a vector-output function
• compute the derivative of a vector-output function
• state and use a geometric interpretation of the derivative of a vector-output function
• understand and use the relations between position, velocity, and acceleration for an object moving on a line, in a plane, or in space
• determine the domain of a function of two variables and plot or describe both level curves and the graph of a simple function of two variables
• determine the domain of a function of three variables and plot or describe level surfaces of a simple function of three variables
• analyze the limit of a function of two or more variables
• analyze continuity function of a function of two or more variables
• use path limits to show that a given limit does not exist for a function of several variables
• compute the partial derivatives of a function of several variables
• read, with understanding, the various notations for partial derivatives
• state and use an appropriate interpretation (as slope or rate of change) of the partial derivatives of a function of several variables
• determine the equation of a tangent plane for a given function at a given point
• determine the linear approximation for a given function
• use the linear approximation to estimate outputs of a function for inputs near a given input
• use the linear approximation to relate percentage changes among related quantities
• use an appropriate chain rule to compute partial derivatives for a given composition of functions
• give a geometric interpretation of a gradient vector
• compute a gradient vector field given a function formula (in cartesian coordinates)
• compute a directional derivative given a function, an input, and a direction
• find and classify (as local minimizer, local maximizer, or neither) all critical points for a given function on a given domain
• find the global minimum and global maximum for a given function on a given domain
• solve applied optimization problems
• state and understand Fubini's Theorem
• set up an iterated integral (in a chosen or specified coordinate system) equal to a double integral for a given function and given domain in the plane
• evaluate a given iterated integral
• set up a double integral that gives the area of a given region of the plane
• set up a double integral or a triple integral that gives the volume of a given region of space
• give a geometric argument for the area element in polar coordinates
• give a geometric argument for the conversion formulas and the volume element for cylindrical coordinates
• give a geometric argument for the conversion formulas and the volume element for spherical coordinates
• set up an iterated integral in a chosen coordinate system (cartesian, cylindrical, spherical) equal to a given triple integral
• plot a given vector field
• set up an interated integral in one variable equal to the line integral of a given scalar function and given curve
• set up an interated integral in two variables equal to the surface integral of a given scalar function and given surface
• compute the total of a quantity given the density of that quantity for a given region
• set up an interated integral in one variable equal to the line integral of a given vector field and given curve
• set up an interated integral in two variables equal to the surface integral of a given vector field and given surface
• compute the divergence of a given vector field
• compute the curl of a given vector field
• give or use a fluid flow interpretation of line integral, surface integral, divergence, and curl for a vector field
• determine if a given vector field has a potential function and, if so, find a potential function
• state and understand the hypotheses and conclusion of the Fundamental Theorem for Line Integrals
• use the Fundamental Theorem for Line Integrals to evaluate the line integral for a given conservative vector field and given endpoints