Course Outline for Mathematics 3
Multivariable Calculus

Effective: Fall 2021
SLO Rev: 12/06/2016
Catalog Description:

MTH 3 - Multivariable Calculus

5.00 Units

Vector valued functions, functions of several variables, partial differentiation, multiple integration, change of variables theorem, scalar and vector fields, gradient, divergence, curl, line integral, surface integral, Theorems of Green, Stokes and Gauss, applications.
Prerequisite: MTH 2.
1701.00 - Mathematics, General
Letter Grade Only
Type Units Inside of Class Hours Outside of Class Hours Total Student Learning Hours
Lecture 5.00 90.00 180.00 270.00
Total 5.00 90.00 180.00 270.00
Measurable Objectives:
Upon completion of this course, the student should be able to:
  1. perform vector algebra in R^3, interpret the results geometrically especially the dot and cross products and apply them to problems in space geometry (eg. orthogonal projection, vector equations of lines and planes);
  2. determine equations of lines and planes;
  3. transform points and equations among rectangular, cylindrical, and spherical coordinates and sketch their graphs as well as quadric surfaces;
  4. sketch the graphs of functions of two variables using level curves, traces in coordinate planes, symmetry, etc.;
  5. parameterize curves using vector functions of one variable and analyze them (eg. find unit tangent, unit normal, curvature);
  6. extend the concepts of limits, continuity, differentiability and differential of single variable functions to functions of two variables;
  7. find the limit of a function at a point;
  8. evaluate derivatives;
  9. determine differentiability;
  10. compute partial derivatives, total differential, gradient, directional derivatives and interpret them geometrically and in terms of rate of change;
  11. write the equation of a tangent plane at a point;
  12. find local extrema and test for saddle points;
  13. apply partial derivatives and/or gradients to problems involving tangent planes and linear approximation, and optimization, especially using Lagrange multipliers;
  14. Solve constraint problems using Lagrange multipliers;
  15. find the divergence and curl of a vector field;
  16. apply differential operators gradient, divergence, curl and Laplacian to scalar and vector fields and interpret the results;
  17. evaluate double and triple integrals directly or using change of variables and explain the geometric interpretation of Jacobians;
  18. compute line integrals using parameterizations for curves;
  19. compute arc length;
  20. parameterize surfaces using vector functions of two variables, and compute their areas;
  21. compute surface integrals of scalar functions and vector functions using parameterization for surfaces;
  22. find scalar potentials for conservative vector field; and
  23. interpret the theorems of Green, Stokes and Gauss (divergence) physically as well as mathematically (as the generalizations of the Fundamental Theorem of Calculus), and apply them to compute line and surface integrals.
Course Content:
  1. Space Analytic Geometry
    1. Rectangular, cylindrical and spherical coordinate systems
      1. Rectangular equation of a plane
    2. Surfaces (cylinders, quadric surfaces and surfaces of revolution)
  2. Parametric Equations
    1. Lines
    2. Planes
  3. Vectors 
    1. Vector operations in R2 and R3
      1. Dot product
      2. Cross product
      3. Triple product
    2. Vector algebra and geometry in R2 nad R3
      1. Lines 
      2. Planes
      3. Projections
      4. Tangent, normal and binormal vectors
    3. Vector-valued functions
      1. Differentiation
      2. Integration
      3. Applications, including velocity and acceleration
  4. Scalar and Vector fields
    1. Vector differential operators
      1. Gradient
      2. Divergence
      3. Curl
      4. Laplacian
      5. Identities involving differential operators
    2. Gradient vector fields
    3. Conservative fields
  5. Real-valued Functions of Several Variables
    1. Level curves
    2. Surfaces
  6. Limits and Continuity
    1. Properties of limits
    2. Properites of continuity
  7. Differentiability and Differentiation
    1. Partial differentiation
    2. Scalar functions of two or more variables
    3. Chain rule
    4. Higher order derivatives
    5. Directional derivatives
    6. Gradient
    7. Optimization
      1. Local and global maxima and minima extrema
      2. Saddle points
      3. Lagrange multipliers
  8. Multiple Integration
    1. Double and triple integrals and their evaluations in rectangular
    2. In polar, cylindrical and spherical coordinates
    3. Change of variables and Jacobians
    4. Path integrals
      1. Integral of scalar functions along a path
      2. Arc length and curvature
      3. Line integrals
      4. Path independence
    5. Surface integral
      1. Parametrically defined surfaces and area
      2. Real-valued functions over surfaces
    6. Applications
      1. Area
      2. Volume
      3. Surface area, 
      4. Center of mass 
      5. Moments of inertia
  9. Vector Analysis
    1. Green's Theorem 
    2. Stokes' Theorem
    3. Scalar potentials and conservative fields
    4. Gauss' Theorem (Divergence Theorem)
Methods of Instruction:
  1. Lecture/Discussion
  2. Demonstration/Exercise
  3. Group Activities
  4. Problem Solving
  5. Distance Education
Assignments and Methods of Evaluating Student Progress:
  1. Find the critical points of f(x,y) = sin x + cos y, and for each, classify as local maximum, local minimum, or saddle point.
  2. Use Lagrange Multipliers to find the maximum and minimum values of f(x, y) = 5x+4y, that occur on the curve x^2+xy+y^2 =28.
  3. Find the average value of f(x,y,z) = sqrt(x^2 + y^2 + z^2) on the solid sphere of radius a, centered at the origin.
  4. Use the general change of coordinates to evaluate the double integral of f(x,y) = (x-y)^3*exp(x-4y) over the parallelogram with vertices (0,0), (1,1), (4,1), and (5,2).
  5. Compute directly the flux of the vector field F(x,y,z) = (x, 2y, 3z) across the closed spherical shell of radius a, centered at the origin. Verify your result by computing the same flux using Gauss’ Divergence Theorem.
  1. Exams/Tests
  2. Quizzes
  3. Home Work
Upon the completion of this course, the student should be able to:
  1. Critically analyze mathematical problems using a logical methodology.
  2. Communicate mathematical ideas, understand definitions, and interpret concepts.
  3. Increase confidence in understanding mathematical concepts, communicating ideas and thinking analytically.
Textbooks (Typical):
  1. Briggs, W., L. Cochran, B. Gillett, E. Schulz (2019). Calculus (3rd). Pearson Education.
  • Access code to MyMathLab (optional)
  • Graphing calculator (optional)
Abbreviated Class Schedule Description:
Vector valued functions, functions of several variables, partial differentiation, multiple integration, Green Theorem, Stokes Theorem, Gauss Theorem and applications.
Prerequisite: MTH 2.
Discipline:
Mathematics*