# Course Summary & Learning Objectives

#### Course Learning Objectives

A student successfully completing this course should be able to:

• LO1 Carry out algorithms and clearly show the steps associated with basic computations.

• LO2 Apply, with guidance, the concepts and computational techniques introduced in this class.

• LO3 Identify geometric and physical content encoded in given mathematical expressions.

• LO4 Analyze problems arising in mathematics, the sciences, and engineering, identify the relevant mathematical concepts and computational techniques, and correctly apply them to find a solution.

• LO5 Identify key similarities and differences between multivariable and vector calculus and calculus of a single variable.

#### Course Summary

Math 275 is an introduction to the geometry of curves and two- and three-dimensional regions, and the calculus of vector-valued functions and functions of more than one variable. This course covers five main topics: vectors and vector operations, the geometry and calculus of curves in two- and three-dimensional space, derivatives of multivariable functions, integration of multivariable functions, and vector calculus. It is important to realize that the material in each topic builds on material introduced earlier in the course, and often generalizes concepts from Calculus I and II.

• Topic 1: Vectors, Vector Operations, and Vector Functions
Vectors are mathematical objects possessing both magnitude (length) and direction. They can be thought of as arrows in space. Vectors are a fundamental tool in applying mathematics to problems occurring in two- and three-dimensional space (and higher dimensions!). In this topic, we learn the mathematical operations that apply to vectors; see how to use vectors and their operations to compute angles and areas, projections, and equations of lines and planes in space; and explore a few basic physical applications of vectors and their operations.

Vector-valued functions can be used to represent curves. Vector functions and their derivatives have many applications. For example, in Calc I and II, the derivative and integral were used to study the position, velocity and acceleration of objects traveling in a straight line. Vector functions can be used to generalize these notions to objects whose paths of travel are curves in two- and three-dimensional space. Position, velocity and acceleration are now represented by vectors.

• Topic 2: Derivatives of Multivariate Functions
In Calc I, we saw that the derivative of a function of a single variable could be interpreted both as an instantaneous rate of change, and as a slope of a tangent line. In this topic, we see that a function of more than one variable has more than one type of derivative. We learn to compute these derivatives; compare their similarities and differences; see how they relate to tangent lines and rates of change; and explore a few of their applications to physical problems, including optimization (max/min) problems. A special vector field, the “gradient” of a function, encodes information about the different types of derivatives of a function.

• Topic 3: Integration of Multivariate Functions
In Calc II, integrals of single-variable function were used to find the area under curves. In this topic, we learn to compute integrals of functions of more than one variable; use double and triple integrals to find planar areas and volumes of regions in space; and see how changing to other coordinate systems (polar, cylindrical and spherical) can be used to simplify integration.

• Topic 4: Vector Calculus
Integration is generalized to functions defined on curves and surfaces in three-dimensional space. We also study vector fields in two- and three-dimensional space, their derivatives (curl and divergence), and integrals of these fields as they act on curves and surfaces in two- and three-dimensional space. Finally, we learn higher-dimensional versions of the Fundamental Theorem of Calculus. This topic brings together all previous topics studied in this class.