In Calculus I and II, we studied functions that accepted a single real number input, , and generated a single real number output, . In this course, we want to generalize the ideas of calculus to functions that accept several inputs and generate several outputs. An example of such a function might be the velocity field of the air around the wing of the airplane in the photo above. We can imagine that at each point in space and at each moment in time, the molecules of air each have their own velocity (speed and direction). The red smoke enables us to visualize and imagine this complicated behavior. Mathematically, the motion of the air is a velocity vector (3 outputs) that depends on position and time (4 inputs). Does calculus apply to such complicated functions?
This course extends the concepts of limits, derivatives, and integrals to vector-valued functions and functions of more than one variable. The topics covered include three-dimensional analytic geometry and vectors, partial derivatives, multiple integrals, line integrals, surface integrals, and the theorems of Green, Gauss (Divergence), and Stokes. Many applications of calculus are included.
Student Learning Outcomes:
Upon successful completion of the course, students will be able to:
- habitually ask questions of mathematical ideas like "What if...?", "How can I visualize this...?", "How is this connected to...?" etc.
- provide context for each idea in the larger scope of the course. E.g. "We're learning about vectors so that we can use vector-valued functions."
- understand vectors algebraically and geometrically and use vectors to represent physical and abstract quantities.
- understand spacecurves and their derivatives geometrically and algebraically.
- apply calculus to compute the arc length, velocity, acceleration and curvature at any point for a space curve.
- evaluate partial derivatives and directional derivatives; understand these derivatives geometrically; find the extrema for functions of several variables.
- construct and evaluate double and triple integrals using rectangular, polar, cylindrical, and spherical coordinate systems as well as change of variables using the Jacobian; apply double and triple integrals to solving geometry and physics problems.
- evaluate line and surface integrals using Green’s Theorem, Stokes' Theorem, and the Divergence Theorem.
- Vectors in two and three dimensions; perform vector operations including dot product, cross product, and projections; find vector and parametric equations of lines and planes, rectangular equation of a plane, and parametric representations of lines and planes; find parametric representations of surfaces
- Vector-valued functions and space curves; determine continuity, determine differentiability; find derivatives, integrals, and arc lengths; analyze the path of a particle using vectors to describe position, velocity, acceleration, speed, curvature, and tangential and normal components of acceleration, binormal vector.
- Functions of several variables, graph functions of several variables; find and graph level curves, surfaces and contour diagrams; find the limit of function at a point; determine continuity; find partial derivatives and differentials, including chain rules, higher-order partial derivatives, gradients, and directional derivatives; determine differentiability; find the equations of tangent planes and normal lines at a point; identify and classify local and global extrema and saddle points.
- Evaluate double integrals and use them to calculate areas and volumes; construct and evaluate double integrals in polar coordinates; use iterated integrals to calculate surface area, evaluate and utilize triple integrals; calculate first and second moments and center of mass; construct and evaluate triple integrals in cylindrical coordinates and spherical coordinates; use the change of variables formula including calculating Jacobians.
- Vector fields; find the gradient vector field and flow of a vector field; evaluate line integrals using parameterizations and the Fundamental Theorem of Line Integrals; apply theorems on independence of path and conservative vector fields; apply Green's Theorem; interpret flux in terms of surface integrals; evaluate surface integrals using parameterizations; find the curl and divergence of a vector field; apply Stokes' Theorem and the Divergence Theorem.
Great news: your textbook for this class is available for free online!
Calculus, Volume 3 from OpenStax, ISBN 1-947172-16-6
You have several options to obtain this book:
- View online (Links to an external site.) (Links to an external site.)
- Download a PDF (Links to an external site.) (Links to an external site.)
You can use whichever formats you want. Web view is recommended -- the responsive design works seamlessly on any device.
Final Grading Scheme
- Any student needing accommodations should inform the instructor. Students with disabilities who may need accommodations for this class are encouraged to notify the instructor and contact the Academic Resource Center (ARC) early in the semester so that reasonable accommodations may be implemented as soon as possible. All information will remain confidential.
- Academic dishonesty and plagiarism will result in a failing grade on the assignment. Using someone else's ideas or phrasing and representing those ideas or phrasing as our own, either on purpose or through carelessness, is a serious offense known as plagiarism. "Ideas or phrasing" includes written or spoken material, from whole papers and paragraphs to sentences, and, indeed, phrases but it also includes statistics, lab results, art work, etc. Please see the student handbook for policies regarding plagiarism.
The syllabus page shows a table-oriented view of the course schedule, and the basics of course grading. You can add any other comments, notes, or thoughts you have about the course structure, course policies or anything else.
To add some comments, click the "Edit" link at the top.