Students will become skilled in computations and applications of infinite sequences and sums. Students will become familiar with the properties of infinite sums to either converge to a finite value, or diverge, possibly to infinity. Students will learn about methods to determine convergence of several important categories of series. Students 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 learn important tools of calculus in higher dimensions. 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 that are defined by multivariate functions, by using partial derivatives and the gradient vector. Students will also learn how to construct approximating tangent planes using partial derivatives. Finally, students will learn how to use the multivariate chain rule to compute derivatives of space curves over surfaces.
Students will also learn methods of multivariable integration on varied 2- and 3D domains using cartesian, polar, and spherical coordinates. Students will be introduced to the tools of integration of multivariate functions over areas and volumes and will learn the use of iterated multiple integration. Similar to single-variable integration, students will learn the technique of multidimensional change-of-variables by utilizing the Jacobian. Specifically, students will learn how to transform between an integral over an area or volume in Cartesian coordinates to one in either polar or spherical coordinates.
Students will become familiar with vector functions that define vector fields in the plane and 3D space. Students will learn the properties of conservative vector fields and their relationship to the fundamental theorem of line integrals.
Students will learn the fundamental vector calculus integral theorems of Green, Stokes', and Divergence. Students will learn meaning and method of computation of the curl and divergence of a vector field, and their use in the aforementioned theorems.
In addition to topical content, students will learn to understand problem descriptions, select the appropriate operations, execute methods accurately, and finally interpret and communicate results.