# MAT-283 Objectives

## MAT283 CALCULUS III

### Course Description:

This course is a continuation of Calculus II (MAT 282) and includes plane curves, parametric equations, vectors, vector-valued functions, tangent and normal vectors, arc-length and curvature, functions of several variables, directional derivatives, gradients, extrema of functions of several variables, Lagrange Multipliers, line integrals, Green’s Theorem, Surface Integrals, the Divergence Theorem, Stokes’ Theorem, and applications to physical sciences and engineering. Prerequisite: Grade of C or better in Calculus II (MAT 282) or Placement.

### Student Learning Outcomes

After completing this course, the students will know and be able to:

1. Demonstrate mastery of mathematical notation and terminology used in this course
2. Demonstrate knowledge of the fundamental principles including laws and theorems relevant to course competencies.
3. Demonstrate a complete understanding pertaining to parametric equations and calculus, such as, slope of tangent line, arc length and area of surface of revolution in parametric form.
4. Demonstrate ability to manipulate vectors.
5. Demonstrate a clear understanding of concepts of velocity and acceleration vectors; unit tangent and normal vectors; arc length and curvature of a curve in space.
6. Demonstrate to execute calculus operations on functions of several variables and their applications.
7. Demonstrate knowledge of underlying concepts of Lagrange multipliers.
8. Demonstrate knowledge to evaluate multiple integrals and their applications.
9. Demonstrate to execute a change of variables for double integrals using the Jacobian.
10. Demonstrate to utilize and evaluate vector fields, line integrals, and their applications.
11. Demonstrate knowledge and complete understanding of Green’s Theorem, Divergence Theorem and Stokes’s Theorem.
12. Demonstrate knowledge, competencies, and thought process to support further study of vector calculus and beyond.

### Instructional Objectives

Unit 1

• Sketch the graph of a curve given by a set of parametric equations.
• Eliminate the parameter in a set of parametric equations.
• Find the slope of a tangent line to a curve given by a set of parametric equations.
• Find the arc length of a curve given by a set of parametric equations.
• Find the surface area of revolution (parametric form).
• Find the area of a region bounded by a polar graph.
• Find the area of a surface of revolution (polar form).

Unit 2

• Write the component form of a vector.
• Perform basic vector operations.
• Write a vector as a linear combination of standard unit vectors.
• Use vectors to solve problems involving force.
• Understand three-dimensional rectangular coordinate system.
• Analyze vectors in space.
• Find dot product.
• Use dot product to find the angle between two vectors.
• Find direction angles.
• Find cross product of two vectors.
• Determine the volume by triple scalar product.
• Write a set of parametric equations for a line in space.
• Write a linear equation to represent a plane in space.
• Find the distances between points, planes, and lines in space.

Unit 3

• Represent a plane curve as a vector-valued function.
• Represent a space curve as a vector-valued function.
• Compute limits of vector-valued functions.
• Compute derivatives of vector-valued functions.
• Compute integrals of vector-valued functions.
• Describe the velocity and acceleration associated with a vector-valued function.
• Use a vector valued function to analyze projectile motion.
• Find a unit tangent vector and a principal unit normal vector.
• Find the tangential and normal components of acceleration.
• Find the arc length of a space curve.
• Use the arc length parameter to describe a plane curve or space.
• Find the curvature of a curve at a point on the curve.

Unit 4

• Find domains of functions of several variables.
• Analyze graphs of functions of two variables.
• Find the level surfaces of functions in two variables.
• Find the level surfaces of functions in three variables.
• Find the limit of functions of several variables.
• Test for continuity of functions of two variables.
• Test the continuity of functions of three variables
• Compute partial derivatives of any order of functions of two or more variables.
• Apply chain rules to compute partial derivatives and total derivatives for functions of several variables.
• Find partial derivatives implicitly.
• Compute gradients of functions of several variables.
• Determine directional derivatives of functions of two or more variables.
• Find directional derivatives and gradients of functions of three variables.
• Determine equations of tangent planes and normal lines to surfaces.
• Find the angle of inclination of a plane in space.
• Find absolute and relative extrema of a function of two variables.
• Use the Second Partial Test to find relative extrema of a function.
• Solve optimization problems involving functions of several variables.
• Understand the Method of Lagrange Multipliers.
• Use Lagrange Multipliers to solve constrained optimization problems.
• Use the Method of Lagrange Multipliers with two constraints.

Unit 5

• Evaluate an iterated integral.
• Use an iterated integral to find the area of a plane region.
• Use a double integral to represent the volume of a solid region and use properties of double integrals.
• Evaluate a double integral as an iterated integral.
• Find the average value of a function over a region.
• Evaluate double and iterated integrals in rectangular and polar coordinates.
• Find the mass of a planar lamina using a double integral.
• Find the center of mass of a planar lamina using double integral.
• Find the moments of inertia using double integrals.
• Apply double integrals to find the volume of a solid region.
• Solve applied problems involving double integrals.
• Use a triple integral to find the volume of a solid region.
• Understand the concept of a Jacobian.
• Use a Jacobian to change variables in a double integral.

Unit 6

• Compute the divergence of a vector field.
• Compute the curl of a vector field.
• Determine whether a vector field is conservative.
• Determine the potential function of a conservative vector field.
• Evaluate a line integral.
• Evaluate a line integral of a vector field.
• Use the Fundamental Theorem of line integrals.
• Determine whether a line integral is independent of path.
• Evaluate line integrals using Green’s Theorem.
• Find a set of parametric equations to represent a surface.
• Evaluate surface integrals.
• Use the Divergence Theorem to calculate flux.
• Solve applied problems using Stokes’ Theorem, such as, the motion of a rotating liquid.