MAT 241 - Calculus III [SUN# MAT 2241]
4 Credits, 4 Contact Hours
4 lecture periods 0 lab periods
Continuation of MAT 231 . Includes vectors in two and three dimensions, vector-valued functions, differentiation and integration of multivariable functions, and calculus of vector fields.
Prerequisite(s): Within the last three years: MAT 231 with a grade of C or better.
Gen-Ed: Meets AGEC - MATH; Meets CTE - M&S.
Course Learning Outcomes
1. Use vector operations to calculate equations of planes and vector equations of lines.
2. Use partial derivatives to analyze rates of change of multivariable functions in a variety of contexts.
3. Evaluate double and triple integrals of multivariable functions in a variety of coordinate systems.
4. Evaluate line and surface integrals in vector fields using a variety of theorems and techniques.
1. Use a Cartesian coordinate system in 3-dimensional space; perform vector operations including the dot and cross products; and find the orthogonal projection onto a vector.
2. Determine equations of lines and planes in space, and identify and classify quadric surfaces.
3. Evaluate limits, derivatives and integrals of vector-valued functions; analyze motion along a curve; and calculate the unit tangent vector, the unit normal vector, and the curvature.
4. Evaluate limits, determine continuity, and calculate partial derivatives of multivariable functions; apply the chain rule and use implicit differentiation; calculate directional derivatives and gradient vectors; find equations of tangent planes; determine extrema and saddle points; and use Lagrange multipliers to find constrained maximum and minimum.
5. Evaluate double integrals in rectangular and polar coordinates; convert between rectangular, cylindrical, and spherical coordinates; evaluate triple integrals in rectangular, cylindrical, and spherical coordinates; and use double and triple integrals to calculate volumes.
6. Determine if a vector field is conservative and find a potential function; evaluate line integrals of real-valued functions and vector fields; evaluate surface integrals of real-valued functions and vector fields; and use Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem to evaluate line integrals and surface integrals.
- Vectors and Analytic Geometry in the Plane and in Space
- Vectors in the plane and in space
- Dot product and cross product
- Orthogonal projections
- Lines, Planes, and Surfaces
- Lines and planes in space
- Quadric surfaces
- Vector Valued Functions
- Graph of a vector valued function
- Parametrized curves
- Arc length
- Unit tangent vector, unit normal vector, and curvature
- Projectile motion
- Functions of Two or More Variables
- Limits and continuity
- Partial derivatives
- Chain rule
- Implicit differentiation
- Linearization and differentials
- Directional derivatives, gradient vectors, and tangent planes
- Local extrema and saddle points
- Absolute extrema
- Lagrange multipliers
- Multiple Integrals
- Double integrals in rectangular and polar coordinates
- Cylindrical and spherical coordinates
- Triple integrals in rectangular, cylindrical, and spherical coordinates
- Calculus of Vector Fields
- Vector fields
- Line integrals
- Path independence, potential functions, and conservative vector fields
- Parametrized surfaces
- Surface area and surface integrals
- Divergence and curl
- Green’s Theorem
- Divergence Theorem and Stokes’ Theorem
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