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Credit- Degree applicable
Effective Quarter: Fall 2020

##### 5 Unit(s)

 (See general education pages for the requirement this course meets.)Requisites: (Not open to students with credit in MATH 1DH.) Prerequisite: MATH 1C or MATH 1CH (with a grade of C or better) or equivalent. Advisory: EWRT 211 and READ 211, or ESL 272 and 273. Hours: Lec Hrs: 60.00 Out of Class Hrs: 120.00 Total Student Learning Hrs: 180.00CAN: Description: Partial derivatives, multiple integrals, vector calculus. Student Learning Outcome Statements (SLO) • Student Learning Outcome: Graphically and analytically synthesize and apply multivariable and vector-valued functions and their derivatives, using correct notation and mathematical precision. • Student Learning Outcome: Use double, triple and line integrals in applications, including Green's Theorem, Stokes' Theorem and Divergence Theorem. • Student Learning Outcome: Synthesize the key concepts of differential, integral and multivariate calculus.

II. Course Objectives

 A. Examine functions of several variables, define and compute limits of functions at points and define and determine continuity
 B. Define and compute partial derivatives, directional derivatives and differentials of multivariable functions and examine conditions of differentiability; find the equation of the tangent plane to a surface at a point
 C. Find local extreme values of functions of several variables, test for saddle points, examine the conditions for the existence of absolute extreme values, solve constraint problems using Lagrange multipliers, and solve related application problems
 D. Use rectangular, cylindrical and spherical coordinates systems to define space curves and surfaces in Cartesian and parametric forms
 E. Integrate functions of several variables
 F. Examine vector fields and define and evaluate line integrals using the Fundamental Theorem of Line Integrals and Green’s Theorem; compute arc length
 G. Define and compute the curl and divergence of vector fields and apply Green’s Theorem, Stokes’s Theorem and the Divergence Theorem to evaluate line integrals, surface integrals and flux integrals

III. Essential Student Materials

 Students will use technology (computers or graphing calculators) to examine mathematical concepts graphically and numerically

IV. Essential College Facilities

 None

V. Expanded Description: Content and Form

 A. Examine functions of several variables, define and compute limits of functions at points and define and determine continuity
 1 Define and find traces parallel to the standard axes, and define and find level curves of functions of several variables
 2 Graph functions of several variables using traces and level curves
 3 Graph and examine level curves and level surfaces of multivariable functions
 4 Examine contour and density graphs of functions of several variables
 5 Model functions of three variables by level surfaces in two variables
 6 Define the limit of a function of several variables
 7 Use the definition of limits to find limits of functions of several variables
 8 Define continuity of functions of several variables and use the definition to evaluate limits of a function at points and determine the continuity of functions over regions in 2D and 3D space
 B. Define and compute partial derivatives, directional derivatives and differentials of multivariable functions and examine conditions of differentiability; find the equation of the tangent plane to a surface at a point
 1 Compute partial derivatives algebraically and numerically
 2 Compute gradients and directional derivatives of functions of two and three variables
 3 Develop the chain rule and find partial derivatives of composite functions of several variables
 4 Find higher order derivatives
 5 Find equations of tangent planes to surfaces
 6 Find equations of normal lines to surfaces
 C. Find local extreme values of functions of several variables, test for saddle points, examine the conditions for the existence of absolute extreme values, solve constraint problems using Lagrange multipliers, and solve related application problems
 1 Find local maxima and local minima of functions of several variables
 2 Use the second derivative test to determine
 a. Local minima
 b. Local maxima
 3 Find absolute extreme values over:
 a. Polygonal boundaries
 b. Curvilinear boundaries
 4 Apply the method of Lagrange Multipliers to find constrained extrema
 5 Solve applied optimization problems from topics such as but not limited to
 a. Analytic and solid geometry
 c. Engineering and physics
 D. Use rectangular, cylindrical and spherical coordinates systems to define space curves and surfaces in Cartesian and parametric forms
 1 Define cylindrical, spherical and rectangular coordinate systems and find equations of selected surfaces and space curves
 2 Find parametric representations for space curves and surfaces using cylindrical, spherical and rectangular coordinates
 3 Apply the chain rule to find ordinary and partial derivatives of space curves and parametric surfaces
 4 Find the equations of tangent planes and normal lines to parametric surfaces
 E. Integrate functions of several variables
 1 Define and evaluate double integrals in rectangular coordinates over rectangles and regions with curvilinear boundaries
 2 Define and evaluate triple integrals in rectangular coordinates over rectangular solids and regions in 3D space with surface boundaries
 3 Evaluate double integrals in polar coordinates over regions with circular and curvilinear boundaries
 4 Use change of variables to evaluate double and triple integrals:
 a. By substitutions
 b. By change of variables and the Jacobian
 5 Solve application problems using double and triple integrals and compute:
 a. Average value of a function
 b. Area, surface area, and volume
 c. Probability
 d. Moments of mass and center of mass
 e. Moments of inertia
 F. Examine vector fields and define and evaluate line integrals using the Fundamental Theorem of Line Integrals and Green’s Theorem; compute arc length
 1 Define vector fields in 2D and 3D space
 2 Produce and examine graphs of vector fields
 a. Find the gradient of a function
 b. Define conservative vector fields and potential functions
 c. Find the generated potential functions of conservative vector fields
 4 Define and evaluate line integrals
 a. Over parametric curves
 b. Of vector fields
 5 Use line integrals to evaluate
 a. Arc length
 b. Work done by a vector field
 6 Apply the Fundamental Theorem of Line Integrals to evaluate line integrals
 7 Apply Green’s Theorem to evaluate line integrals
 8 Examine path dependence/independence of line integrals of vector fields
 9 Solve problems related to conservative and non-conservative vector fields
 G. Define and compute the curl and divergence of vector fields and apply Green’s Theorem, Stokes’s Theorem and the Divergence Theorem to evaluate line integrals, surface integrals and flux integrals
 1 Define and find the curl of a vector field and interpret geometrically
 2 Define and find the divergence of a vector field and interpret geometrically
 3 Apply Green’s Theorem in vector form to evaluate line integrals
 4 Find parametric representations of surfaces and determine their orientation
 5 Define and compute the surface area of parametric surfaces
 6 Define and evaluate surface integrals of real-valued functions including parametrically defined surfaces
 7 Define and evaluate flux integrals over surfaces in rectangular coordinates and parametric surfaces
 8 Apply Stokes’s Theorem to evaluate surface and line integrals
 9 Apply the Divergence Theorem to evaluate surface and triple integrals

VI. Assignments

 B. Problem solving exercises
 C. A selection of homework, quizzes, group projects and exploratory worksheets
 D. Optional project synthesizing various concepts and skills from course content

VII. Methods of Instruction

 Lecture and visual aids Discussion of assigned reading Discussion and problem-solving performed in class In-class exploration of internet sites Quiz and examination review performed in class Homework and extended projects Guest speakers Collaborative learning and small group exercises Collaborative projects Problem solving and exploration activities using applications software Problem solving and exploration activities using courseware

VIII. Methods of Evaluating Objectives

 A. Periodic quizzes and/or assignments from sources related to the topics listed in the curriculum will be evaluated for completion and accuracy to assess students’ comprehension and ability to communicate the course content as well as to provide timely feedback to students on their progress.
 B. At least three one-hour exams without projects or at least two one-hour exams with projects are required. Students are expected to provide complete and accurate solutions to problems that include both theory and applications by integrating methods and techniques studied in the course. Students shall receive timely feedback on their progress.
 C. One two-hour, comprehensive, final examination is required, in which students are expected to display comprehension of course content and be able to choose methods and techniques appropriate to the various types of problems that cover course content. The final examination may be in a variety of formats such as free response, numerical problem solving, essay and short answer, objective-type and/or multiple choice.
 D. Projects (optional) Projects may be used to enhance students' understanding of topics studied in the course in group or individual formats, whether communicating their understanding orally through classroom presentation or in writing. Evaluation is to be based on completion and comprehension of course content, and students shall receive timely feedback on their progress.

IX. Texts and Supporting References

 A. Examples of Primary Texts and References
 1 James Stewart, "Calculus: Early Transcendentals", 8th Ed. Brooks/Cole 2015.
 B. Examples of Supporting Texts and References
 1 Hass, Weir, and Thomas, "University Calculus, Early Transcendentals", 3rd Ed. Pearson 2015.
 2 Anton, Bivens, and Davis, "Calculus: Early Transcendentals", 10th Ed. Wiley 2011.
 3 Hughes-Hallett, et al., "Calculus: Single and Multivariable", 6th Ed. Wiley 2012.