The content, assignments, and assessments for Calculus III are aligned to the following learning outcomes. A full list of course learning outcomes can be viewed here: Calculus III Learning Outcomes.
Module 1: Describing curves through parametric equations and polar coordinates
- Identify parametric equations
- Apply calculus to parametric equations
- Understand polar coordinates and their application
- Determine area and arc length in polar coordinates
- Distinguish properties of parabolas, ellipses, and hyperbolas
Module 2: Interpret functions of two or three independent variables in multidimensional space
- Understand vectors and their operations
- Apply vectors to three-dimensional space
- Use the dot product
- Use the cross product
- Interpret equations of lines and planes in space
- Distinguish properties of cylinders, ellipsoids, paraboloids, and hyperboloids
- Convert coordinates between rectangular and nonrectangular coordinates
Module 3: Interpret vector-valued functions to determine the velocity, acceleration, arc length, and curvature of an object’s trajectory
- Apply vector-valued functions to curves in the plane and in three-dimensional space
- Apply calculus to vector-valued functions
- Determine arc length and curvature in space
- Describe motion in space
Module 4: Develop methods to solve differential equations of functions with several variables
- Interpret functions of several variables
- Determine limits and continuity of functions of several variables
- Calculate the derivatives of functions of several variables
- Apply tangent planes to three-dimensional surfaces
- Apply the chain rule to functions of several variables
- Calculate directional derivatives
- Identify extrema and critical points for a function of two variables
- Apply Lagrange multipliers to solve optimization problems
Module 5: Apply integration techniques to functions containing more than one variable and other coordinate systems
- Calculate double integrals over rectangular regions
- Apply double integrals to general regions
- Use double integrals on polar rectangular regions
- Calculate triple integrals over three-dimensional space
- Evaluate triple integrals using cylinderical and spherical coordinates
- Use triple integrals to locate centers of mass and moments of inertia
- Calculate mutliple integrals using a change of variables
Module 6: Generalize vector fields and its application to Green’s, Stokes’ and the divergence theorems
- Identify vector fields
- Calculate line integrals along curves
- Explain conservative vector fields
- Apply Green’s theorem
- Determine divergence and curl for a vector field
- Interpret surface integrals
- Use Stokes’ theorem
- Apply the Divergence theorem
Module 7: Develop methods to solve second-order differential equations
- Interpret second-order linear equations
- Solve nonhomogeneous linear equations
- Apply second-order differential equations to real-world concepts
- Use power series to solve differential equations
