# 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.