MTH 265 Calculus III (4 cr.)
Prepares students for further study in calculus. Focuses on extending the concepts of function, limit, continuity, derivative, integral, and vector from the plane to the three dimensional space. Topics include vector functions, multivariate functions, partial derivatives, multiple integrals, and an introduction to vector calculus. Designed for mathematical, physical, and engineering science programs. This course replaces MTH 277 or MTH 178 and is the third course in a three-course sequence. Prerequisite: Placement in MTH 265 or completion of MTH 264 or equivalent with a grade of C or better. Lecture 4 hours per week.
Major Topics:
Vectors and the Geometry of Space
Vector Functions
Partial Derivatives
Multiple Integrals
Vector Calculus
Objectives and Skills:
Identify and apply the parts of the three-dimensional 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 quadratic surfaces and cylinders using the concepts of trace and cross-section
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
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 and interpret the gradient, and use it to find directional derivative
Apply knowledge of the concepts of extrema for functions of several variables; Lagrange multipliers
Define and evaluate double integral by the midpoint rule and describe its simplest properties
Calculate iterated integrals by Fubini’s Theorem
Calculate double integrals and use geometric interpretation of double integral as a volume to calculate such volumes; compute 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 define the simplest properties; 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
Describe vector fields in two and three dimensions graphically; determine if vector fields are conservative
Identify the meaning and 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