Required Prerequisite Knowledge: To succeed in this course a student must have good mastery of precalculus as well as first-year differential and integral calculus.
This course is offered via the Internet and via distributed DVD's in an asynchronous mode. The student will receive instructional information and assignments via these modes and will respond to assignments by submitting work through web forms.
The student must have standard access to the Internet and must have the ability to access the content on the DVD's. The material on the DVD's is accessible using a variety of media players (e.g., Windows Media Player).
The instructor is available via web forms (to which students will be introduced at the very beginning of the course), and will normally respond by the end of the day following your submission (and more typically on the same day) with answers to properly posed questions, feedback on your efforts, and other information. Exceptions may occur in the event of Internet problems or other technical events.
Broad goals and Purpose of the CourseThe student will gain proficiency with scalar and vector functions of two or more variables and the calculus of these functions, including but not limited to the ability to:
Each assigned task and problem constitutes a specific objective, which is to complete that problem or task and understand as fully as possible its relationship to the stated goals of the assignment and to other concepts, problems and situations encountered in the course.
Unless you are the 1 person in 1000 with the combination of aptitude and learning style capable of doing it otherwise, solving problems is the only way to learn the material in this course.
Ability to perform the tasks listed below provide a good foundation for solving the assigned problems, the process of which leads to mastery.
Chapter 9: Vectors and Geometry in Space
9.1 - 9.2 9.3
- Interpret geometrically the results of vector addition, subtraction and scalar multiplication.
- Relate {initial point, final point, component form, linear combination of standard unit vectors}
- Relate displacement, force or velocity in vector form and use to solve associated problems.
9.4
- Relate {v_1, v_2, v_1 dot v_2, theta, || v_1 ||, || v_2 ||, proj_v_2 ( v_1 ) , proj_v_2 ( v_1 ), u_1 = v_1 / || v_1 ||, u_2 = v_2 / || v_2 ||}
- Relate { v, direction cosines of v }
- Relate { `ds, F_ave, `dW } on an interval..
9.5
- Relate { v, w, v X w, theta, sin(theta), direction of v X w, plane containing v and w }
- Relate { u dot (v X w), geometric properties of parallelepiped defined by u and v and w }
9.6
- Sketch a 3-dimensional coordinate system and the plane through a given point on a given coordinate axis of the plane perpendicular to that axis.
- Relate { (x_1, y_1, z_1) and (x_2, y_2, z_2) and line through points, v = v_0 + t u, x = x_0 + a t and y = y_0 + b t and z = z_0 + c t, (x - x_0) / a = (y - y_0) / b = (z - z_0) / c, parametric equations for the line defined by the points, u = a i + b j + c k , v_0 = x_0 i + y_0 j + z_0 k}
- Relate { (x_0, y_0, z_0), a i + b j + c k, a (x - x_0) + b ( y - y_0 ) + c ( z - z_0) = 0, a x + b y + c z = d, equation of plane, vector normal to plane, (x_1, y_1, z_1) in plane, (x_2, y_2, z_2) in plane, linearly independent vectors v and w parallel to plane, x- y- and z-axis intercepts of plane, sketch of plane}
- Relate {point 1 = (x_1, y_1, z_1), point 2 = (x_2, y_2, z_2), r = distance between point 1 and point 2, locus of points at distance r from (x_1, y_1, z_1), locus of points equidistant from point 1 and point 2, equation of plane bisecting line segment from point 1 to point 2, equation of sphere of radius r centered at (x_1, y_1, z_1)}
9.7
- Relate {equation of circle or ellipse in xy or xz or yz plane, equation of cylinder with axis through center of and perpendicular to plane of circle or ellipse, sketch of cylinder}
- Relate {equation of quadric surface, equation and/or sketch (in two or three dimensions as specified) of intersection of quadric surface with given plane parallel to specified coordinate plane, equation and/or sketch of family of equations of intersections of quadric surface with family of planes parallel to specified coordinate plane, topographic map of quadric surface relative to specified coordinate plane, sketch of quadric surface}
- ... relate and execute?
- Recognize and write equations for quadratic surfaces.
- Recognize and write equations for surfaces of revolution.
- Represent surfaces in space using cylindrical coordinates.
- Represent surfaces in space using spherical coordinates.
Chapter 10: Vector-Valued Functions
10.1 10.2
- Form new functions as requested, by a combination of addition and scalar multiplication, elements of a set of scalars, scalar functions and vector functions.
- Analyze and sketch a space curve given by a vector-valued function.
- Extend the concepts and properties of limits and continuity to vector-valued functions.
- Determine intervals of continuity for a given vector-valued function.
10.3
- Determine intervals on which a given vector-valued function is smooth.
- Apply the definition of the derivative to validate properties of vector derivatives.
- Calculate the derivative of a given vector function expression (i.e., an expression built using a combination of addition, scalar multiplication, dot products and cross products).
- Integrate a given vector-valued function.
- Given a vector-valued function modeling the position of a particle as a function of time, calculate the vector functions representing speed, velocity, acceleration and direction of motion.
- Given the vector-valued function modeling the velocity of a particle as a function of time and its position at a given time, calculate its position function.
10.4
- Describe the velocity and acceleration associated with a given vector-valued function representing the position of a particle vs. time.
- Analyze projectile motion using a vector-valued function.
- Analyze the motion of a satellite or planet using polar coordinates, with radial and tangential unit vectors.
10.5
- Determine the unit tangent vector at a point on a given space curve.
- For a given vector function describing a curve, calculate the unit tangent and principal unit normal vectors.
- Find the tangential and normal components of acceleration of a particle whose position is given by a vector-valued function.
- Calculate the distance traveled along a given space curve between to given points.
- Describe a plane curve or space curve using the arc length parameter.
- Given position as a function of arc distance along a space curve, find the unit tangent and principal unit normal vectors as functions of arc distance.
- Give and explain the meaning of the definition of curvature.
- Calculate the curvature at a given point of a space curve given parametrically.
- Given position as a vector function of time, find the vector functions for the tangential and normal components of the acceleration function.
- Using the tangential and normal components of the acceleration function for an appropriate vector function model, analyze the motion of an object in a real-world situation.
Chapter 11: Functions of Several Variables
11.1 11.2
- Demonstrate understanding of the notation for a function of several variables.
- Sketch the graph of a function of two variables.
- Sketch level curves for a function of two variables.
- Sketch level surfaces for a function of three variables.
11.3
- Define a neighborhood in the plane and demonstrate understanding of the meaning of the definition and the terms used in the definition.
- Define the limit of a function of two or more variables.
- Evaluate the limit of a given function of two or more variables at a given point.
- Extend the concept of continuity to a function of two or more variables.
- Test a given function of two or more variables for continuity at a given point.
11.4
- Explain, aided by a sketch, the geometric meaning of a partial derivative of a function of two variables.
- Calculate the partial derivative of a given function of two or more variables, with respect to a specified variable.
- Find higher-order and mixed partial derivatives of a function of two or three variables.
11.5
- Explain how the vector normal to a surface defined by a function of two variables, at a given point of the surface, is related to the partial derivatives of the function at that point.
- Explain how the vector normal to a surface at a given point can be used, along with the coordinates of that point, to determine whether a third point is on the associated tangent plane.
- Calculate incremental approximations using the tangent plane.
- Explain how incremental approximations and differentials are related.
- Analyze the behavior of a function of two or more variables within a given neighborhood of a point, using the total differential.
- Calculate incremental approximations using the total differential.
11.6
- Explain the chain rule for a function of two variables, where each variable is in turn a function of the same single parameter, in terms of rates of change.
- Use the Chain Rules to calculate derivatives of a given function of several variables, where the variables themselves are functions of a set of parameters.
- Calculate partial derivatives implicitly.
11.7
- Explain, aided by a sketch, how the directional derivative in a given direction, at a given point, is related to the partial derivatives at that point of a given function of two variables.
- Explain how the directional derivative in a given direction, at a given point, is related to the plane tangent at that point to the graph of a given function of two variables.
- Calculate the directional derivative of a given function of two or more variables in the direction of a given vector.
- Calculate the gradient of a given function of two or more variables.
- Given the gradient of a function at a point, and a vector, find the directional derivative in the direction of that vector.
- Analyze the behavior of a given function of two or more variables, as revealed by the maximal property of the gradient.
- Analyze the behavior of a given function of two or more variables, as revealed by the normal property of the gradient.
- Calculate the normal vector to a given level surface at a given point.
- Calculate the equations of the tangent plane and the normal line for a level surface, at a given point on the surface.
- Analyze an applied situation which can be modeled by a function of two or more variables, using the directional derivative and the gradient.
11.9
- Apply critical points and the Second Partials Test to find relative extrema and saddle points of a function of two variables.
- Find absolute and extrema of a function of two variables within a given region.
- By minimizing an appropriate function of two variables, find the least squares regression line for a given set of y vs. x data.
- 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.
- Find equations of tangent planes and normal lines to surfaces.
- Find the angle of inclination of a plane in space.
- Compare the gradients Ñ f(x, y) and Ñ f(x, y, z).
Chapter 12: Multiple Integration
12.1 12.2
- Given a statement of Fubini's Theorem, explain it in terms of sketches.
- Evaluate a given iterated integral.
- Calculate the area of a rectangular plane region using an iterated integral.
12.3
- Given a function defined on a specified region of the plane, use double integrals to integrate the function over that region.
- Given a region of the plane, describe it in rectangular coordinates in terms of partitions and limits.
- Given a region of the plane and a function defined over that region, describe the region and the integral of the function over the region in rectangular coordinates in terms of partitions, sample points and limits.
- Given a definite double integral, express it as an equivalent integral with the order of integration reversed.
- Use a double integral to represent the area of a given plane region.
- Use a double integral to represent the volume of a given solid region.
12.4
- Define and explain the meaning of the area increment in polar coordinates.
- Given a region of the plane, describe it in polar coordinates in terms of partitions and limits.
- Given a region of the plane and a function defined over that region, describe the region and the integral of the function over the region in polar coordinates in terms of partitions and limits.
- Write and evaluate double integrals in polar coordinates.
- Relate the Jacobian of the transformation from rectangular to polar coordinates to the area increment in polar coordinates.
- Express the area of a given plane region as a double integral in polar coordinates.
- Express the volume of a given solid region as a double integral in polar coordinates.
12.5
- Given a function of two variables and a point of the plane, find the normal vector at the corresponding point of the graph, and using the normal vector calculate the corresponding ratio of surface area to area in the plane (more specifically, the ratio of the area of a small neighborhood of the surface defined by the graph to the area of that region's projection into the plane).
- Given a function of two variables and a region of the plane, describe the surface area of the portion of the graph whose projection is the plane region, in terms of partitions, sample points, normal vectors, area ratios and limits.
- Given a function of two variables interpreted as a density function, and a region of the plane, describe the mass of the region in terms of partitions, sample points, and limits.
- Calculate the mass of a planar lamina using a double integral.
- Calculate the area of a surface defined parametrically.
- Given a function defined on a specified region of space, use triple integrals to integrate the function over that region.
- Given a region of space, describe it in rectangular coordinates in terms of partitions and limits.
- Given a region of space and a function defined over that region, describe the region and the integral of the function over the region in rectangular coordinates in terms of partitions, sample points and limits.
- Given a definite triple integral, express it as an equivalent integral with a different specified order of integration.
- Use a triple integral to represent the volume of a given solid region.
12.6
12.7
- Given a region of space and a density function defined over that region, describe the region and the mass within that region in rectangular coordinates in terms of partitions, sample points and limits.
- Given a region of space, a density function defined over that region and an axis, describe the region and its moment or moment of inertia in terms of partitions, sample points and limits.
- Calculate the center of mass of a given solid region, given a density function and an axis.
- Calculate the moment of inertia of a given solid region, given a density function, about a given axis.
12.8
- Given a region of space defined in rectangular, cylindrical or spherical coordinates, describe it in terms of either or both of the other two, as specified.
- Given an expression in rectangular, cylindrical or spherical coordinates, express it in terms of either or both of the other two, as specified.
- Given a region of space, describe it in cylindrical coordinates in terms of partitions and limits.
- Given a region of space and a function defined over that region, describe the region and the integral of the function over the region in cylindrical coordinates in terms of partitions, sample points and limits.
- Given a region of space, describe it in spherical coordinates in terms of partitions and limits.
- Given a region of space and a function defined over that region, describe the region and the integral of the function over the region in spherical coordinates in terms of partitions, sample points and limits.
- Using cylindrical coordinates express as triple integrals the mass and moment of inertia about a given coordinate axis of a given region with given density function.
- Using spherical coordinates express as triple integrals the mass and moment of inertia about a given coordinate axis of a given region with given density function.
- Using cylindrical coordinates express as a triple integral the center of mass of a given region with given density function.
- Using spherical coordinates express as a triple integral the center of mass of a given region with given density function.
- Using the Jacobian, make a specified change of variable for a multiple integral.
- Choose an appropriate change of variable and use it to simplify the region of integration and/or the integrand of a given multiple integral.
Chapter 13: Vector Analysis
13.1
- Given a vector function of two variables, sketch the vector field defined by this function, and the flow lines associated with the field.
- Sketch examples of vector field representing circular flow and convergent or divergent flow.
- Sketch examples of central force fields and the associated equipotentials.
- Given a set of equipotentials sketch the associated force field.
- Given a vector field calculate its divergence and curl.
- Given a scalar function of two variables determine whether it is harmonic.
.13.2
- Given a scalar function and a curve defined parametrically in the plane or in space, partition a given interval of the parameter and define the line integral in terms of the partition, sample points and limits.
- Given a scalar function and a curve defined parametrically in the plane or in space, write and evaluate the line integral of the function over a specified interval of the curve.
- Given a vector function and a curve defined parametrically in the plane or in space, partition a given interval of the parameter and define the line integral of the tangential component of the field along the curve in terms of the partition, sample points and limits.
- Given a vector field and a curve in space or in the plane, write and evaluate the line integral of the tangential component of the field along the curve.
- Given a curve in space or in the plane and a function defining the mass density, with respect to length, as a function of position find the mass of the physical object represented by the curve.
- Given a vector function defining a force field and a path, in the plane or in space, determine the work performed as the object moves along the path.
13.3
13.4
- Determine whether the line integral of a vector field, between two given points of the plane or space, is path independent.
- Given a path in the plane or in space, and a scalar function f, find the line integral of the gradient of f between two given points.
- Determine whether a given vector field is conservative.
- Given a conservative vector field calculate a scalar potential function for that field.
- Use the curl to determine if a given vector field is conservative.
- Apply the various equivalent conditions for path independence to solve problems and analyze real-world situations.
- Relate path independence to conservation of energy.
13.5
- Explain how Green's Theorem applies to a small rectangular region.
- Explain how a grid can be used to prove Green's Theorem for a standard region.
- Use Green's Theorem to express a line integral around a closed curve as a double integral over the region bounded by the curve, and vice versa.
- Use Green's Theorem to express the area of a region bounded by a closed curve as a line integral.
- Apply Green's Theorem to doubly-connected regions.
- Apply Green's Theorem to a region containing a singular point.
- Use Green's Theorem to relate the line integral of the tangential component of a vector field to a double integral involving the curl of the field.
- Use Green's Theorem to relate the line integral of the normal component of a vector field to a double integral involving the divergence of the field.
13.6
- Determine the orientation of a given surface.
- Given a function of two variables defining a surface, a scalar function defined in space and a region of the plane, describe in terms of partitions, sample points, normal vectors, area ratios and limits the surface integral of the scalar function on the portion of the surface whose projection is the plane region.
- Given a surface in space and a scalar function defined in a region of space which contains the surface, write and evaluate the double integral representing the integral of the function over the surface.
- Apply a surface integral to calculate the mass of a laminar region in space, given the appropriate density function.
- Given a surface in space and a vector function defined in a region of space which contains the surface, write and evaluate the double integral representing the flux of the field through the surface.
- Find a set of parametric equations to represent a surface.
- Find a normal vector and a tangent plane to a parametric surface.
- Compute surface integrals for surfaces which are defined parametrically.
13.7
- Use Stokes' Theorem to relate a flux integral over a surface to a line integral around the boundary of the surface.
- Verify Stokes' Theorem for a given vector field and a given surface.
- Use Stokes' Theorem to evaluate a surface integral by evaluating a line integral.
- Explain Stokes' Theorem in terms of the cumulative tendency of a fluid to swirl across a surface and the circulation of the fluid around the boundary of that surface.
- Use the Divergence Theorem to relate the flux through a closed surface to the integral of the divergence over a volume.
- Given a vector function and a region of space, verify the Divergence Theorem.
- Apply the Divergence Theorem to evaluate a surface integral over an open surface.
- Apply the Divergence Theorem to applications involving fluid flow and heat flow.
- Use the Divergence Theorem to calculate the flux of a given vector field through a given closed surface.
Specific objectives are also stated at the course homepage, where they are correlated assignment by assignment.
Requirement of communicationRegular communication is required of the student. This includes turning in assignments in a timely fashion and responding in a timely manner to feedback on these assignments. Any deviation of more than three days from the chosen schedule of the course must be approved in advance by the instructor. Exceptions will of course be made in the event of documented illness or other unexpected emergencies, but the instructor should be informed of such situations within a reasonable time of occurrence.
The Blackboard site for your course directs you to the Initial Information document and instructions for establishing communication, information on the nature of your course and the Orientation and Startup instructions you will need to understand and navigate the course. This process will constitute appropriately the first week's assignments for your course (about the first half of the week during the shorter summer term), and will show you the basic navigation of the website including how to communicate, submit work, locate assignments and due dates, and more.
The text is specified in Textbook Information, which the student will have encountered prior to arriving at this page. Any student who has not noted Textbook Information is advised to review all information to be sure no other essential details have been missed.
Units to be covered:
Chapters 6-11 inclusive, plus supplementary material posted by instructor.
Chapter Topics:
Chapter 9: Vectors in the Plane and in Space
Chapter 10: Vector-valued Functions
Chapter 11: Partial Differentiation
Chapter 12: Multiple Integration
Chapter 13: Vector Analysis
Specific information regarding assignments and areas covered is included on the homepage.
Instructional methodsStudents will complete and submit the assignments specified on the homepage.
The instructor will respond in a timely fashion to any work submitted, making suggestions where improvement is needed and posing questions designed to enhance the student's learning experience. The student will be required to respond to all critiques, except those designated otherwise.
Questions posed by students and the instructor's responses will be posted to a site, specified in at the beginning of the course, for the student's review.
Students may on occasion be asked to critique work done by other students. Full student anonymity will be preserved, with no reference to the identity of any party in this exchange.
The instructor is available via web forms (to which you will be introduced at the very beginning of the course), and will normally respond by the end of the day following your submission (and more typically on the same day) with answers to properly posed questions, feedback on your efforts, and other information. Exceptions may occur in the event of Internet problems or other technical events.
Use of email: Prior to registration and receipt of initial instructions students my use Email to communicate with the instructor. However email is much less reliable than web forms, and after registration and receipt of initial instructions anything sent through email should first be sent using the appropriate form.
Grading policyFive tests and will be administered, one over each chapter.
A student's portfolio, consisting of instructor responses to assigned work and/or daily quizzes, will at the end of the term be assigned a grade. A student who completes all assigned work in the prescribed manner can expect to make an A on this aspect of the course. The average of grades assigned on this work will count as 1/4 of a test grade. If this average is higher than the average on other tests, it will be counted as 1/2 of a test grade.
Raw test scores will be normalized to the following scale, according to the difficulty of the test, as specified in advance of each test by the instructor:
A: 90 - 100
B: 80 - 90
C: 70 - 80
D: 60 - 70
F: Less than 60.
The final grade will be a weighted average according to the above guidelines. A summary of the weighting is as follows:
Test #1: Weight .5 or 1.0, to the advantage of the student
Test #2: Weight 1.0
Test #3: Weight 1.0
Test #4: Weight 1.0
Test #5: Weight 1.0
Assignment/Quiz Grade Average: Weight .25 or .5, to the advantage of the student.
The table below summarized the calculation of course grades:
assessment weighting contribution to total score test 1 1 test score * 1 test 2 1 test score * 1 test 3 1 test score * 1 test 4 1 test score * 1 test 5 1 test score * 1 portfolio 1/4 <= p_weight <= 1/2 portfolio score * p_weight total of weightings total of contributions Final average = total of contributions / total of weightings
Criteria for Grading of Tests:
Tests will consist of problems designed to measure the level of your achievement of the course goals.
Each problem is graded on a 10-point scale, with the following guidelines:
In the event of a college-wide emergency
In the event of a College-wide emergency, course requirements, classes, deadlines, and grading schemes are subject to changes that may include alternative delivery methods, alternative methods of interaction with the instructor, class materials, and/or classmates, a revised attendance policy, and a revised semester calendar and/or grading scheme.
In the case of a College-wide emergency, please refer to the following about changes in this course:
Instructor’s email dsmith@vhcc.edu (however, you should use your access page for the most reliable responses)
For more general information about the emergency situation, please refer to:
· Web site - www.vhcc.edu
· Telephone Number - 276-739-2400
· Emergency Text Messaging or Phone System- Virginia Highlands Community College uses VHCC Alert to immediately contact you during a major crisis or emergency. VHCC Alert delivers important emergency alerts, notifications and updates to you on your E-mail account (work, home, other), cell phone, pager or smartphone/PDA (BlackBerry, Treo & other handhelds). VHCC Alert is a free service offered by VHCC. Your wireless carrier may charge you a fee to receive messages on your wireless device. VHCC will test the alert system each semester. Register online at alert.vhcc.edu or by sending a text message to 411911 keyword: VHCC
In the event of an emergency just regarding this class, the instructor will contact all students via email, and may post information to your access site. You should check both email and your access site.