Course Goals
To study the nature and significance of calculus for students of science, technology, engineering, and mathematics disciplines.
To demonstrate various applications of calculus to problems from the social sciences, physical sciences, and engineering.
To present an exposition of calculus that incorporates graphical, numerical, and algebraic analysis, without undue emphasis on theoretical abstraction or routine mechanical manipulation.
To use technology to illustrate calculus concepts and verify calculus solutions to application problems.
To provide students with an exposure to the logical reasoning of mathematics.
Core Course Topics
Vectors and Vector -Valued Functions
Perform vector addition and scalar multiplication of vectors algebraically and geometrically.
Identify and solve problems involving vectors in the plane and in 3-space both geometrically and algebraically.
Compute the dot product for vectors in 2 and 3 dimensions and the cross product for vectors in 3 dimensions.
Use the dot and cross products in applications such as computing the angle between 2 vectors, vector projections, scalar components, work and torque.
Construct vector-valued functions in order to describe lines and curves in 3-space.
Determine limits of vector-valued functions using numerical, algebraic and graphical methods.
Determine continuity or discontinuity of vector-valued functions.
Determine derivatives and integrals of vector-valued functions.
Find the vector components of velocity and acceleration for particle motion in 3-space.
Find the length of a curve in 3-space and be able to parameterize a given curve with respect to arc length.
For a given curve, determine the unit tangent, unit normal, unit binormal and curvature.
Functions of Several Variables
Determine the equation of a plane in 3-space using both the vector form and the scalar form.
Write the equation for a given quadratic surface.
Draw accurate and useful renditions of a surface given an equation by hand and using appropriate graphing application software.
Use graphs of level curves (contour maps) to obtain information about an arbitrary surface.
Determine limits of functions of several variables using numerical, algebraic and graphical methods.
Determine continuity or discontinuity of functions of several variables.
Find the first and second partial derivatives of functions using differentiation rules.
Determine the gradient and directional derivatives for a function of several variables.
Use the appropriate chain rule to differentiate composite functions of several variables.
Use the gradient and directional derivatives to find instantaneous rates of change and related rates of change in application problems.
Use the gradient and directional derivatives and the Second Partials Test to find properties of functions of several variables: maxima and minima and saddle points, and use this information to sketch the graph of functions.
Solve optimization problems using the method of Lagrange Multipliers.
Use the gradient and tangent plane to determine the linear approximation of a function of several variables.
Integration of Functions of Several Variables
Integrate functions of 2 variables over regions in the xy-plane using Riemann sums and iterated integrals with respect to rectangular and/or polar coordinates.
Integrate functions of 3 variables over 3-dimensional regions using Riemann sums and iterated integrals with respect to rectangular, cylindrical, or spherical coordinates.
Use multiple integrals to solve application problems involving areas, volumes, masses, first and second moments, centers of mass, and moments of inertia.
Vector Fields
Sketch vector fields by hand.
Interpret vector fields as models of fluid flow.
Compute line integrals over curves in 2- or 3-space.
Determine whether a vector field is conservative or not.
Use the Fundamental Theorem for Line Integrals to compute line integrals under appropriate conditions.
Use Green’s Theorem in the plane in flux form or in circulation form to compute line integrals.
Compute the 2-dimensional divergence and curl for a vector field in 2-dimensions.
Use line integrals to solve application problems involving work, mass, circulation and heat flux.
Compute the 3-dimensional divergence and curl for a vector field in 3-dimensions.
Compute surface integrals of scalar-valued functions.
Compute surface integrals of vector fields.
Use Stokes’ Theorem to evaluate line and surface integrals.
Use the Divergence Theorem to calculate flux.
Upon successful completion of this course, students will be able to: