MTH 261 - Analytic Geometry and Calculus III

1. Demonstrate an understanding of the concept of a vector and manipulate and represent vectors geometrically and algebraically.
   1. Perform basic calculations with vectors such as addition, subtraction, scalar multiplication, and finding the magnitude of a vector.
   2. Find the cross product and dot product of two vectors and use them in various applications.
   3. Describe some key differences between the dot product and the cross product.
   4. Find the angle between two given vectors.
   5. Use a dot product to calculate the work done by a constant force.
   6. Use a cross product to calculate torsion.
   7. Use dot products to find the projection of one vector onto another.
   8. Find the equation of a plane.
   9. Find the parametric equation of a line in space.
2. Develop an understanding of the relationships between numerical, graphical, and algebraic representations of curves and surfaces in space.
   1. Graph standard quadric surfaces and curves in space.
   2. Recognize the relationships between curves, surfaces and their equations.
   3. Graph using cylindrical and spherical coordinates.
   4. Convert among rectangular, cylindrical, and spherical coordinates.
3. Define, apply, and identify several properties of vector-valued functions.
   1. Identify differences between vector-valued functions and scalar-valued functions.
   2. Evaluate a limit of a vector-valued function.
   3. Evaluate a derivative of a vector-valued function.
   4. Use and understand the velocity and acceleration of vector-valued functions.
   5. Construct a TNB frame.
   6. Calculate and apply the curvature and the torsion of a curve in space.
4. Define, apply, calculate with, and identify properties of a multivariable real valued function.
   1. Determine the domain and range of a function.
   2. Construct level curves and level surfaces of a function
   3. Calculate the limit of functions when they exist.
   4. Use paths on a surface to show when a limit of a function does not exist at a point.
   5. Determine if a function is continuous at a point.
   6. Calculate partial derivatives of a function.
   7. Calculate the linear approximations to a function.
   8. Use partial derivatives to find absolute and local extrema and saddle points for two variable scalar functions.
   9. Use Lagrange multipliers to find extrema for constrained functions.
   10. Define, calculate and apply the gradient of a function.
   11. Use the gradient of a function to calculate directional derivatives.
5. Apply, evaluate, and understand integrals of multi-variable scalar-valued functions.
   1. Define double and triple integrals.
   2. Construct a region of integration.
   3. Represent areas and volumes with double and triple integrals.
   4. Evaluate double and triple integrals using rectangular coordinates.
   5. Calculate surface area.
   6. Evaluate surface integrals.
   7. Use cylindrical and spherical coordinates to evaluate triple integrals.
   8. Change the order or variables of integration when appropriate.
   9. Apply formulas that deal with mass, center of mass, and moments.
6. Develop an understanding of vector fields.
   1. Define a vector field.
   2. Give examples of vector fields in an abstract and physical setting.
   3. Evaluate line integrals in conservative and nonconservative fields.
   4. Explain the relationships between conservative fields, path independence, and potential functions.
   5. Calculate work in a variety of contexts.
   6. Explain and evaluate the curl and divergence of vector fields.
   7. Explain and apply Green’s Theorem.
   8. Apply Stokes’ and Gauss’ Theorems and explain their relationship to Green’s Theorem.
7. Communicate effectively about mathematics.
   1. Verbally describe solutions to problems using appropriate terminology.
   2. Provide complete written explanations of concepts using appropriate terminology.
8. Use technology appropriately to do mathematics.