Students will be able to construct and evaluate integrals covering a wide range of applications, including how to compute of an arc lengths of curves, the average value of a function, the center of mass for objects, and the computation of energy as a force integrated over a distance.
Students will understand how to compute infinite sums of geometric type. They will be able to represent functions as a Taylor series, and use Taylor's theorem to approximate functions and estimate error from using finitely many terms of the Taylor series.
Students will become familiar with 2- and 3-dimensional coordinate systems, vectors and vector operations including the dot and cross product, and equations of lines, planes, and other surfaces. Students will also learn how to represent motion of objects in 3D using vector functions, how to represent velocity and acceleration using vector projections into tangential and centripetal coordinates of acceleration, and how to characterize curves in space by computing arc length and curvature.
Students will be able to characterize aspects of surfaces and volumes using partial derivatives and the gradient vector, tangent planes, and the chain rule.
Students will learn to compute multivariable integrals on varied 2D domains using cartesian and polar coordinates, and 3D domains using cartesian and spherical coordinates. Students will also be able to use multivariable integrals to compute areas of surfaces.
Students will learn to characterize properties of vector fields using various operators, and compute path and flux integrals in 2- and 3D.
Students will learn to use the major theorems of vector calculus: Green's and Stokes', and the Divergence theorem related to path and flux integrals.
In addition to topical content, students will also gain further mastery in problem solving fluency: Students will be able to read and interpret problem objectives, be able to select and execute appropriate methods to achieve the aforementioned objectives, and be able to interpret and communicate results.