MTH 265 Calculus III
Course Title: MTH 265 Calculus III
Course Description
Focuses on extending the concepts of function, limit, continuity, derivative, integral and vector from the plane to the three dimensional space. Covers topics including vector functions, multivariate functions, partial derivatives, multiple integrals and an introduction to vector calculus. Features instruction designed for mathematical, physical and engineering science programs. Lecture 3 hours per week. 3 credits.
General Course Purpose
The general purpose of this third course in a three course sequence is to prepare students for further study in mathematics, engineering and science programs by providing the necessary competencies in calculus concepts in the three dimensional space.
Course Prerequisites/Corequisites
Prerequisite: Completion of MTH 264 or equivalent with a grade of C or better.
Course Objectives
Upon completing the course, the student will be able to:
Vectors and the Geometry of Space

Identify and apply the parts of the threedimensional coordinate system, distance formula and the equation of the sphere

Compute the magnitude, scalar multiple of a vector, and find a unit vector in the direction of a given vector

Calculate the sum, difference, and linear combination of vectors

Calculate the dot product and cross product of vectors, use the products to calculate the angle between two vectors, and to determine whether vectors are perpendicular or parallel

Determine the scalar and vector projections

Write the equations of lines and planes in space

Draw various quadric surfaces and cylinders using the concepts of trace and crosssection
Vectors and the Geometry of Space

Sketch vector valued functions

Determine the relation between these functions and the parametric representations of space curves

Compute the limit, derivative, and integral of a vector valued function

Calculate the arc length of a curve and its curvature; identify the unit tangent, unit normal and binormal vectors

Calculate the tangential and normal components of a vector

Describe motion in space
Partial Derivatives

Define functions of several variables and know the concepts of dependent variable, independent variables, domain and range.

Calculate limits of functions in two variables or prove that a limit does not exist;

Test the continuity of functions of several variables;

Calculate partial derivatives and interpret them geometrically, calculate higher partial derivatives

Determine the equation of a tangent plane to a surface; calculate the change in a function by linearization and by differentials,

Determine total and partial derivatives using chain rules,

Calculate directional derivatives and interpret the results

Identify the gradient, interpret the gradient, and use it to find directional derivative

Apply intuitive knowledge of concepts of extrema for functions of several variables, and apply them to mathematical and applied problems. Lagrange multipliers.
Multiple Integrals

Define double integral, evaluate a double integral by the definition and the midpoint rule and describe the simplest properties of them.

Calculate iterated integrals by Fubini’s Theorem

Calculate double integrals over general regions and use geometric interpretation of double integral as a volume to calculate such volumes. Some applications of double integrals may include computing mass, electric charge, center of mass and moment of inertia

Evaluate double integrals in polar coordinates to calculate polar areas, evaluate Cartesian double integrals of a particular form by transforming to polar double integrals

Define triple integrals, evaluate triple integrals, and know the simplest properties of them. Calculate volumes by triple integrals

Transform between Cartesian, cylindrical, and spherical coordinate systems; evaluate triple integrals in all three coordinate systems; make a change of variables using the Jacobian
Vector Calculus

Describe vector fields in two and three dimensions graphically; determine if vector fields are conservative, directly and using theorems

Identify the meaning and setup of line integrals and evaluate line integrals

Apply the connection between the concepts of conservative force field, independence of path, the existence of potentials, and the fundamental theorem for line integrals. Calculate the work done by a force as a line integral

Apply Green's theorem to evaluate line integrals as double integrals and conversely

Calculate and interpret the curl, gradient, and the divergence of a vector field

Evaluate a surface integral. Understand the concept of flux of a vector field

State and use Stokes Theorem

State and use the Divergence Theorem
Major Topics to be Included
Vectors and the Geometry of Space
Vector Functions
Partial Derivatives
Multiple Integrals
Vector Calculus