Analytic Geometry and Calculus III

MTH 261

Course Description

Prerequisite: MTH 162 with a grade of "C" or better. Includes solid analytical geometry, vectors, partial differentiation, multiple integration, line and surface integrals, Green's, Stokes', and Gauss’ theorems. A CAS GRAPHING CALCULATOR IS REQUIRED. (60-0)

Outcomes and Objectives

Demonstrate an understanding of the concept of a vector and manipulate and represent vectors geometrically and algebraically.

Objectives:

• Perform basic calculations with vectors such as addition, subtraction, scalar multiplication, and finding the magnitude of a vector.
• Find the cross product and dot product of two vectors and use them in various applications.
• Describe some key differences between the dot product and the cross product.
• Find the angle between two given vectors.
• Use a dot product to calculate the work done by a constant force.
• Use a cross product to calculate torsion.
• Use dot products to find the projection of one vector onto another.
• Find the equation of a plane.
• Find the parametric equation of a line in space.

Develop an understanding of the relationships between numerical, graphical, and algebraic representations of curves and surfaces in space.

Objectives:

• Graph standard quadric surfaces and curves in space.
• Recognize the relationships between curves, surfaces and their equations.
• Graph using cylindrical and spherical coordinates.
• Convert among rectangular, cylindrical, and spherical coordinates.

Define, apply, and identify several properties of vector-valued functions.

Objectives:

• Identify differences between vector-valued functions and scalar-valued functions.
• Evaluate a limit of a vector-valued function.
• Evaluate a derivative of a vector-valued function.
• Use and understand the velocity and acceleration of vector-valued functions.
• Construct a TNB frame.
• Calculate and apply the curvature and the torsion of a curve in space.

Define, apply, calculate with, and identify properties of a multivariable real valued function.

Objectives:

• Determine the domain and range of a function.
• Construct level curves and level surfaces of a function
• Calculate the limit of functions when they exist.
• Use paths on a surface to show when a limit of a function does not exist at a point.
• Determine if a function is continuous at a point.
• Calculate partial derivatives of a function.
• Calculate the linear approximations to a function.
• Use partial derivatives to find absolute and local extrema and saddle points for two variable scalar functions.
• Use Lagrange multipliers to find extrema for constrained functions.
• Define, calculate and apply the gradient of a function.
• Use the gradient of a function to calculate directional derivatives.

Apply, evaluate, and understand integrals of multi-variable scalar-valued functions.

Objectives:

• Define double and triple integrals.
• Construct a region of integration.
• Represent areas and volumes with double and triple integrals.
• Evaluate double and triple integrals using rectangular coordinates.
• Calculate surface area.
• Evaluate surface integrals.
• Use cylindrical and spherical coordinates to evaluate triple integrals.
• Change the order or variables of integration when appropriate.
• Apply formulas that deal with mass, center of mass, and moments.

Develop an understanding of vector fields.

Objectives:

• Define a vector field.
• Give examples of vector fields in an abstract and physical setting.
• Evaluate line integrals in conservative and nonconservative fields.
• Explain the relationships between conservative fields, path independence, and potential functions.
• Calculate work in a variety of contexts.
• Explain and evaluate the curl and divergence of vector fields.
• Explain and apply Green’s Theorem.
• Apply Stokes’ and Gauss’ Theorems and explain their relationship to Green’s Theorem.