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Analytic Geometry and Calculus III

MTH 261

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.

Communicate effectively about mathematics.

Objectives:

  • Verbally describe solutions to problems using appropriate terminology.
  • Provide complete written explanations of concepts using appropriate terminology.

Use technology appropriately to do mathematics.