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Math 50C --- Multivariable Calculus

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Catalog Description

The third in the series of three calculus courses. Multivariable Calculus applies the techniques and theory of differentiation and integration to vector-valued functions and functions of more than one variable. The course presents a thorough study of vectors in two and three dimensions, vector-valued functions, curves and surfaces, motion in two and three dimensions, and an introduction to vector fields.

Special notes or advisories: Extensive computer visualization is an integral component of this course.

Prerequisites

Math 50B (or equivalent) with a grade of "C" or better

Describe representative skills without which the student would be highly unlikely to succeed: Students must be well-versed in the fundamental theorems of differential and integral calculus, including the ability to differentiate and integrate standard functions of one variable.

Recommended Preparation

Course Learning Outcomes

What should the student be able to do as a result of taking this course? State some of the objectives in terms of specific, measurable student accomplishments.

1. Read, write, and speak accurately about mathematical ideas and use correct mathematical notation.
2. Use graphing technology to visualize curves and surfaces in two and three dimensions, explore mathematical concepts, and verify their work.
3. Use the theory of vectors as a fundamental problem-solving tool.
4. Differentiate and integrate vector-valued functions and functions of more than one variable.
5. Use the properties of curves to analyze the motion of objects in two and three dimensions.
6. Apply the mathematics of multivariate functions to real-world problems and applications.
7. Use numerical, graphical, symbolic, and verbal representations to solve problems and communicate with others.

Course Content

• Themes: What themes, if any, are threaded throughout the learning experiences in this course?
  • Problem solving
  • Writing
  • Technology
  • Communication
• Concepts: What concepts do students need to understand to demonstrate course outcomes?
  • The use of mathematical software as a fundamental problem-solving tool, particularly in visualizing the curves and surfaces of multivariable calculus.
  • The presentation of mathematical solutions in a logical coherent structure, including the use of fundamental writing skills, grammar, and punctuation.
  • Vectors.
  • Differentiation.
  • Integration.
  • The connections between integration and differentiation.
  • Curves and Surfaces.
  • Vector fields.
• Issues: What primary issues or problems, if any, must students understand to achieve course outcomes.
  • The appropriate use of technology in the problem-solving process.
  • The connection between mathematics and the real world.
  • The role of the student in becoming a successful learner.
  • The recognition that the problem-solving skills learned in this class are applicable in future mathematics classes and classes in related fields, such as physics, engineering, etc.
• Skills: What skills must students master to demonstrate course outcomes?
  1. Vectors in two and three dimensions
  2. • Use vector notation, perform vector operations, and interpret the results geometrically in two and three dimensions.
     • Compute the length and direction of a vector in two and three dimensions.
     • Use the unit vector as a fundamental problem-solving tool.
     • Compute and apply the dot and cross products, and interpret the results geometrically.
     • Use the vector concept to determine the equations of lines and planes.
  3. Planar and space curves described by vector-valued functions of one variable
  4. • Draw the graph of the function, sometimes manually, but more often with assistance from mathematical software.
     • Compute limits, derivatives, and integrals of vector-valued functions.
     • Compute arc length.
     • Find intersections and collision points.
     • Compute unit tangent and normal vectors, curvature, and interpret the results geometrically.
     • Compute the position, velocity, speed, and acceleration of a particle in the plane or in space.
  5. Surfaces
  6. • Use mathematical software to assist in the drawing of surfaces that are:
       • defined explicitly by a function of two variables,
       • defined implicitly by an equation in three variables, or
       • defined parametrically as a vector function of two variables, including but not limited to, cylindrical and spherical coordinates.
     • Develop manual technique to assist with the drawing of the standard quadric surfaces. Verify results with the use of mathematical visualization software.
  7. Differentiation of functions of several variables
  8. • Compute limits and partial derivatives, and interpret the results geometrically.
     • Find and draw the tangent plane to a surface, and show how the result can be used to approximate the function near the point of tangency.
     • Use the chain rule to find the derivative of a composition of two or more functions.
     • Compute directional derivatives and the gradient, and interpret the results geometrically.
     • Use mathematical software to draw and illustrate the relation between the gradient field and the level curves (surfaces) of a function of two (three) variables.
     • Apply differentiation theory to optimize functions of two variables.
  9. Integration of functions of several variables
  10. • Compute double and triple integrals as iterated integrals, and interpret the results geometrically.
      • Compute double and triple integrals using polar, cylindrical, and spherical coordinates, and interpret the results geometrically.
      • Use multiple integration to determine volume, surface area, mass, and the centroid or center of mass of objects in two and three dimensions.
  11. Vector fields
  12. • Use mathematical software to draw representations of vector fields.
      • Use a line integral to compute the work done on a particle by a vector field as the particle moves along a path in the vector field.
      • Use the graphical representation of a vector field and the usual tests to determine if a vector field is conservative, and if so, calculate a line integral by finding the potential function and using the fundamental theorem for line integrals.
      • Calculate the work or circulation around a closed curve in the plane using Green's Theorem.
      • Calculate the flux across the boundary of a simple planar region using Green's Theorem.
      • Time permitting:
        • Calculate the curl and divergence of a vector field, and interpret the results geometrically and physically.
        • Calculate the surface integral of a function over a parametrized surface.
        • Calculate the work or circulation around a closed curve in space using Stokes' Theorem.
        • Calculate the flux across the boundary of a simple solid region using the Divergence Theorem.

Representative Learning Activities

What will the students be doing (i.e., Listening to lectures, participating in discussions and/or group activities, attending a field trip, etc.)? Relate the activities directly to the Course Learning Outcomes.

• Listening to lectures.
• Participating in group activities and/or assignments.
• Participating in class assignments and/or discussions.
• Completing homework assignments.
• Using mathematical software to complete activities designed to foster a deeper level of understanding of the concepts and skills developed in this class.

Assessment Tasks

How will the student show evidence of achieving the Course Learning Outcomes? Indicate which assessments (if any) are required for all sections.

• Representative assessment tasks:
• • In-class examinations and/or quizzes.
  • Homework assignments.
  • Take-home examinations and/or quizzes allow the instructor to include questions and/or exercises that require the use of the computer and mathematical software to supplement the analysis. Extra time allows the students to develop their writing and presentation skills.
  • Writing assignments designed to develop communication of mathematical concepts.
  • Group and/or individual projects and presentations.
• Required assessments for all sections – to include but not limited to:
• • Homework assignments, including computer activities that deepen the level of understanding.
  • Student projects and/or examinations.

Examples of Appropriate Texts or Other Readings

• Author: Stewart; Title: Calculus, Early Transcendentals, Sixth Edition; Date: 2007

Last Revision: 9/28/07