Multivariable Calculus Common Course Outline

Course Information Organization South Central College Developers Thomas Henry Course Number MATH 233 Division Liberal Arts and Sciences Department Mathematics Total Credits 4

Description Multivariable Calculus extends the notions of Calculus I and Calculus II to functions of more than one variable. Topics include such things as curves and surfaces in Euclidean n-space, partial derivatives, directional derivatives, tangent planes and differentials, double- and triple-integrals, the rectangular, cylindrical and spherical coordinate systems, line integrals, surface integrals, Green's theorem, Stokes' theorem and the divergence theorem. Credits: (3 Lecture/1 Lab) (MNTC Goal Area 4); (Prerequisites: MATH 132 with a grade of C or better)

Prerequisites MATH 132, with a grade of C or better.

Competencies 1. Describe objects in the three-dimensional rectangular coordinate system Learning Objectives a. Interpret equations and inequalities geometrically b. Graph equations of certain three-dimensional objects c. Derive the equation of a sphere 2. Explain two- and three-dimensional vector operations Learning Objectives a. Define vectors via components b. Compute the magnitude of a vector c. Define vector addition and subtraction d. Define scalar multiplication e. Derive properties of vector operations 3. Apply the dot product to vector operations Learning Objectives a. Define dot product algebraically b. Extend this to a trigonometric interpretation c. Establish the criterion for orthogonal vectors d. Prove various properties of the dot product e. Write a vector as a sum of orthogonal vectors f. Explain work by a constant force in terms of the dot product 4. Apply the cross product to vector operations Learning Objectives a. Prove various properties of the cross product b. Review determinants c. Calculate triple scalar products 5. Describe lines and planes in space Learning Objectives a. Give the vector equation for a line b. Give parametric equations for a line c. Find the distance from a point to a line in space d. Derive the equation of a plane e. Find the distance from a point to a plane 6. Describe cylinders and quadric surfaces Learning Objectives a. Derive the equation of the parabolic cylinder b. Derive the equation of an ellipsoid c. Derive the equation of an hyperboloid d. Derive the equation of certain cones e. Derive the equations of certain paraboloids f. Identify saddle points 7. Explain the behavior of vector-valued functions Learning Objectives a. Define a vector-valued function b. Define the limit of a vector-valued function c. Define continuity of a vector-valued function d. Define the derivative of a vector-valued function e. Interpret velocity, direction, speed and acceleration in terms of vector-valued functions 8. Extend the calculus to vector-valued functions Learning Objectives a. Derive differentiation rules for vector-valued functions b. Define the indefinite integral of a vector-valued function c. Define the definite integral of a vector-valued function 9. Interpret length of a smooth curve Learning Objectives a. Compute the length of arc b. Compute speed on a smooth curve c. Define the unit tangent vector 10. Explain curvature Learning Objectives a. Define curvature precisely b. Derive a formula for computing curvature c. Define the principal unit normal vector d. Derive a formula for computing the unit normal vector e. Extend curvature and normal vectors to space curves f. Define torsion 11. Explore real-valued functions of more than one variable Learning Objectives a. Define a function of n independent variables b. Explain interior and boundary points c. Explain open and closed sets d. Explain bounded and unbounded regions in the plane 12. Graph functions of more than one variable Learning Objectives a. Depict a surface algebraically b. Illustrate graph behavior by means of level curves c. Extend b, above, to the notion of level surface d. Define interior and boundary points for space regions 13. Extend the ideas of limits and continuity to higher dimensions Learning Objectives a. Define the limit of a function of two variables b. Intuit properties of limits of functions of two variables c. Define continuity for functions of two variables 14. Define partial derivative Learning Objectives a. Define the partial with respect to x b. Define the partial with respect to y c. Extend these ideas to functions of more than two variables d. Explain the connection between partial derivatives and continuity e. Explore Clairaut's Theorem f. State the connection between differentiability and continuity 15. Prove the Chain Rule for partial derivatives Learning Objectives a. Derive the Chain Rule for functions of two variables b. Derive the Chain Rule for functions of three variables c. Apply the Chain Rule to physical problems 16. Implement the directional derivative Learning Objectives a. Define the directional derivative b. Interpret the directional derivative c. Compute the directional derivative d. Define the gradient vector e. Illustrate algebraic rules of the gradient vector f. Show the directional derivative is a dot product g. Explore properties of the directional derivative 17. Determine tangent planes Learning Objectives a. Define tangent plane b. Define normal line c. Find the plane tangent to a point d. Find the line normal to a point e. Find the plane tangent to a surface 18. Linearize a function Learning Objectives a. Use the differential to estimate change in a direction b. Explain linearization and the standard linear approximation c. Define the total differential 19. Find extreme points Learning Objectives a. Define local maximum b. Define local minimum c. Give the first derivative test for local extreme values d. Define critical point e. Define saddle point f. Explain the second derivative test for local extreme values g. Extend the above to the problem of absolute extrema on closed bounded regions 20. Develop the properties of double integrals Learning Objectives a. Render a double integral over a rectangular region b. Interpret a double integral as a volume c. Calculate a double integral via Fubini's Theorem d. Compute a double integral over certain bounded non-rectangular regions e. Find limits of integration 21. Apply double integrals to real-world problems Learning Objectives a. Find the area of a bounded region in the plane b. Find the average value of a function c. Find the moment of a thin flat plate d. Find the center of mass of a thin flat plate 22. Interpret double integrals in polar form Learning Objectives a. Describe the double integral in polar form b. Find the limits of integration c. Determine area in polar coordinates d. Relate polar and Cartesian integrals 23. Extend the calculus to triple integrals in rectangular coordinates Learning Objectives a. Define volume as a triple integral b. Find the limits of integration c. Find the average value of a function in space d. Compute a mass in three dimensions e. Compute a moment in three dimensions 24. Calculate triple integrals in other coordinate systems Learning Objectives a. Integrate in cylindrical coordinates b. Relate rectangular to cylindrical coordinates c. Describe the spherical coordinate system d. Relate rectangular to spherical coordinates e. Integrate in spherical coordinates f. Convert between the various coordinate systems 25. Define the line integral over a curve Learning Objectives a. Evaluate a line integral b. Explore useful properties of line integrals c. Interpret mass, moment and other properties in terms of line integrals 26. Depict vector fields Learning Objectives a. Define a gradient field b. Define work done over a smooth curve c. Evaluate a work integral 27. Explore special instances of vector fields Learning Objectives a. Define path independence b. Prove the Fundamental Theorem of Line Integrals c. Apply a and b, above, to problems involving work 28. Explain exact differential forms Learning Objectives a. Define exact differential form b. Give the component test for exactness c. Show how to prove that a differential form is exact 29. Apply Green's Theorem in the Plane Learning Objectives a. Define divergence b. Define the k-component of curl c. Explain the normal form of Green's Theorem d. Explain the tangential form of Green's Theorem e. Prove Green's theorem for certain special regions 30. Solve problems involving surface integrals Learning Objectives a. Explain the formula for surface area as a double integral b. Define surface integral c. Define surface area differential d. Define flux e. Find mass and moments of thin shells 31. Parameterize a surface Learning Objectives a. Parameterize various common surfaces b. Define a smooth parameterized surface c. Define area of a smooth surface d. Explain a parametric surface integral 32. Apply Stokes' Theorem Learning Objectives a. Explain Stokes' Theorem b. Prove Stokes' Theorem for certain surfaces c. Show the relationship of curl and grad 33. Apply the Divergence Theorem Learning Objectives a. Define divergence of a vector field in three dimensions b. Prove the Divergence Theorem for certain regions