Math 251 : Multivariable and Vector Calculus
Office: 125C, Engineering 1
Phone: (312) 567-3128
E-mail: kaul [at]
Time: 1:50pm, Tuesday-Thursday-Friday.
Place: 124 Engineering 1 Bldg.
Office Hours: 11:30am-12:30pm Tuesday, 12:30-1:30pm Thursday, and by appointment. Emailed questions are also encouraged.
Tutoring Service: Mathematics tutoring at the Academic Resource Center.
Online Problem Practice: Calculus III at WebAssign and/ or at COW (Calculus on Web).
The Course Information Handout has extensive description of the course - topics, textbook, student evaluation policy, as well as other relevant information. Read it carefully!
The official course syllabus.
Advice for students:
How to read Mathematics? The Basics, More Details.
How to study and learn Math.
Understanding Mathematics - a study guide
You need to be familiar with the material from Calculus I and II, and earlier.
Use the following to review and remind yourselves of all that you already should know:
Paul Dawkin's Algebra Review Sheet
Paul Dawkin's Trignonometry Review Sheet
Paul Dawkin's Calculus I and II Review Sheet
- Tuesday, 1/17 : Dates for Mid-term Exams #1 and #2 and Final Exam have been announced. Look below in the appropriate section.
- Tuesday, 1/10 : Check this webpage regularly for homework assignments, announcements, etc.
- Exam #1 : February 24th, Friday. Syllabus: All topics, examples and problems corresponding to HW#1 to HW#5 (lectures from Jan 10th to Feb 10th).
- Exam #2 : April 6th, Friday. Syllabus: All topics, examples and problems corresponding to HW#6 to HW#9 (lectures from Feb 14th to March 16th).
- Final Exam : May 3rd, Thursday, 2pm to 4pm. Topics: Everything covered in class during the semester.
Homework will be assigned on WebAssign every week, mostly on Friday.
It will typically be due next week on Thursday, followed by a quiz on Friday.
You will be required to set up your own account using the "class key" that I will give you in class. Go to WebAssign. Click on the Log In link on the right hand side of the page. Click on the I Have A Class Key link. The Class Key is composed of three fields. In the first field you will enter the word iit (lower case) followed by two four digit numbers unique to your section. Click Submit. If the information about your section is correct, click Yes, This Is My Class. At this point, follow the instructions to create a username and password. Once your account is set up, you will not have to use the Class Key again.
You will be given a grace period of approximately 14 days from the first day of classes to access WebAssign before you need to purchase your account online.
Important Advice for Homework:
You should first attempt each assigned problem on paper like regular paper and pen assignments (log off the WebAssign website). Save this written work for review before quizzes and exams. and in case you have to discuss your solution with me. Instructor cannot help you if you don't record your work carefully on paper when you need help from him.
Carefully read the WebAssign Student Guide to understand how to format your solutions.
When you have finished entering your answers, click on "Submit" and WebAssign will immediately grade your solutions. You can then re-work your incorrect problems and re-submit them to improve your score. You can submit your assignment up to 5 times without penalty.
After two incorrect attempts, look at the book again, review your notes and re-do the problem from scratch. Do not guess an answer. Remember you have only 5 attempts for each problem, so do not waste your attempts on frivolous trial and error solutions.
Do not enter an equivalent expression when your answer is marked wrong. x*(2*x-5) is the same as (2*x-5)*x, changing from one form to another will not change the validity of your answer. The software behind Webassign matches equivalent expressions, so do not send an email to your instructor that "WebAssign is marking your correct answer wrong". Its extremely rare for such errors to occur.
Avoid simple mistakes like: wrong syntax; incorrect usage of parentheses, x*(2*x-5) is not the same as x*2*x-5; using the incorrect letter for a parameter or a variable in a symbolic solution.
If you have technical difficulties with the WebAssign website, please contact their technical support directly online.
Homework Solutions will be visible soon after the due time. Use these solutions to prepare for the in-class quiz on Friday (next day). Especially note the way of writing and describing the solutions. I use the same descriptive details when I solve examples in class, so pay attention in class also.
For the quizzes and exams, you need to be able to describe the intermediate steps/ calculations/ substitutions for the solution. Just writing the final answer without justifying it will lead to deduction of points.
- Tuesday, 1/10 : Discussion of course aim, HW and quiz, etc. 3-D geometry, coordinate axes & planes, notation and distance formula for points in the space, equations of planes parallel to one of the coordinate planes, examples for description of surfaces and bodies in the space, Handout examples for surfaces and its solutions , equation of sphere, vectors - formula for displacement vector. (From Sections 13.1 and 13.2)
- Thursday, 1/12 : vectors - algebraic and geometric perspectives of addition, scalar multiplication and difference, length of a vector, Handout examples for vector addition and usage in plane geometry, properties of vector addition and scalar multiplication, unit vectors, i,j,k vectors. (From Sections 13.2 and 13.3)
- Friday, 1/13 : Handout for surfaces and its solutions distributed in class. Unit vectors, unit vector in direction of a, dot product of two vectors - properties, geometric interpretation, angle between vectors, orthogonality, sign and value of dot product with changing angle, scalar and vector projections; Handout for 13.3 and discussion of its solutions. (From Section 13.3)
- Tuesday, 1/17 : Work as dot product, 2 by 2 and 3 by 3 Determinants, algebraic definition of cross product of two vectors, orthogonality and right-hand-rule, length of the cross product vector in terms of the angle and its relation to area of a parallelogram, condition for two vectors to be parallel, examples, cross product of basis vectors, non-properties and properties of cross product. (From Sections 13.3 and 13.4)
- Thursday, 1/19 : non-properties and properties of cross product, scalar triple product, etc., torque as a cross product; Handout for cross-product problems and discussion of its problems, Algebraic method for solving vector cross product equations; Vector, parametric, and symmetric equations of a line in 3-D space and the relation between the three definitions. (From Sections 13.4 and 13.5)
- Friday, 1/20 : Quiz#1 and the discussion of its solutions. Vector, parametric, and symmetric equations of a line in 3-D space, Equation of line passing through two points, Equation of line segment passing through two points, Intersections of a line with planes and axes, skew lines with an example. (From Section 13.5)
- Tuesday, 1/24 : Vector, scalar, and linear equations of a plane in 3-D space and the relation between the three definitions, plane passing through a point and two vectors, plane passing through three points; examples for equations of lines and planes satisfying certain conditions, Angle between two planes, Equation of line of intersection of two planes, Distance from a point to a plane, distance between two parallel planes. (From Section 13.5)
- Thursday, 1/26 : Cylinder in 3-D: circular, parabolic and hyperbolic, Quadric surfaces, Method sketching and identifying curves using 2-D cross-sections, examples - parabolic cylinder, circular cylinder, ellipsoid, hyperboloid, etc. Examples and discussion of Handout for section 13.6. (From Section 13.6)
- Friday, 1/27 : Quiz#2 and the discussion of its solution. Distance between two parallel or skew lines; Vector-valued functions- domain, limit, and continuity. (From Sections 13.5 and 14.1)
- Tuesday, 1/31 : Vector-valued functions- limit, and continuity, Relation between vector functions and parametric equation of a curve in 3-D space, examples; Vector functions Handout and its discussion, Derivative of a vector function and its relation to the component functions, Tangent vector and Tangent line - their relation, unit tangent vector, Examples. (From Sections 14.1 and 14.2)
- Thursday, 2/2 : Differentiation rules for vector functions, definite and indefinite integrals of vector functions, Arc length of a space curve, Arc length function and its relation to the original vector function using FTOC, re-parametrization of a curve w.r.t. the arc length. (From Sections 14.2 and 14.3)
- Friday, 2/3 : Quiz#3 and the discussion of its solution. Curvature - definition in terms of arc length, alternate formula in terms of the original parameter, Handout for Ch. 14 and its discussion. (From Section 14.3)
- Tuesday, 2/7 : Completion of the discussion of Handout for Ch. 14, another formula for curvature using second-order derivative and cross product, curvature for plane curves, examples; normal vector, Functions of two variables - domain, range, and graphs with examples. (From Sections 14.3 and 15.1)
- Thursday, 2/9 : Level curves- handout and discussion, functions of 3 or more variables, level surfaces; limit of a 2-variable function - distance versus paths, Non-existence of limits - condition and examples. (From Sections 15.1 and 15.2)
- Friday, 2/10 : Quiz#4 and discussion of its solution. Non-existence of limits - examples with non-linear paths, Methods for existence of limit - substitution in terms of distance, substitution in terms of polar coordinates, Limit laws and squeeze theorem for 2-variable functions. (From Section 15.2)
- Tuesday, 2/14 : Continuous functions, Polynomial and rational functions and their continuity, Composition of continuous functions, short intro to corresponding concepts for 3 or more variable functions, Partial derivatives at a point and as a function with examples. (From Sections 15.2 and 15.3)
- Thursday, 2/16 : Implicit partial differentiation, Geometric interpretation of partial derivative, higher order partial derivatives, Clairut's Theorem, An example of a PDE, Tangent Plane to a surface given by a 2-variable function and its relation to the partial derivatives. (From Sections 15.3 and 15.4)
- Friday, 2/17 : Quiz#5 and discussion of its solution, Linearization and linear approximation, Definition of being differentiable and its relation to linearization, Sufficient condition on first order partial derivatives that makes a function differentiable. (From Section 15.4)
- Tuesday, 2/21 : Differential for 2 or more variable functions and its relation to change of value & error estimate, Chain rule I and II and general chain rule with examples for multivariable functions whose variables are defined in terms of multiple parameters, examples. (From Section 15.4 and 15.5)
- Thursday, 2/23 : Review session for Midterm Exam #1.
- Friday, 2/24 : Midterm Exam #1.
- Tuesday, 2/28 : Example for General chain rule, Formulas for implicit differentiation (2 and 3 variables), Directional derivatives - definition and formula using gradient, Gradient of a function - generalization to 3-variables, direction and value of maximum rate of change. (From Sections 15.5 and 15.6)
- Thursday, 3/1 : Distribution and discussion of Mid-term Exam #1 with solutions and score distribution. Gradient of a function - generalization to 3-variables, direction and value of maximum rate of change (Method of steepest ascent/ descent) using gradient, Formula for a tangent plane to a level surface of a 3-variable function, Formula for a tangent plane to a level curve of a 2-variable function. (From Sections 15.6 and 15.7)
- Friday, 3/2 : Quiz# 6 and discussion of its solutions, Equation of the Normal line to a surface at a given point, local maximum and minimum of 2-variable functions, critical points and examples of local max/min problems. (From Sections 15.6 and 15.7)
- Tuesday, 3/6 : Second derivative test and examples of optimization problems with 3 variables changed into 2 variable max/min problems, Extreme value theorem and extreme points on the boundary, example for finding global max and min of a continuous function in a closed and bounded set D. (From Sections 15.6 and 15.7)
- Thursday, 3/8 : example for finding global max and min of a continuous function in a closed and bounded set D, Lagrange multipliers for 2- and 3-variable functions with one or two constraint functions, Method of Lagrange Multipliers and its application to find global max and min of a continuous in a closed and bounded set D, Example for Method of Lagrange Multipliers for two constraints. (From Section 15.8)
- Friday, 3/9 : Quiz# 7 and discussion of its solution, Examples for finding local max/min that illustrate how to find all the possible critical points and what to do when SDT does not apply. Review of Riemann sum (from Calc I). (From elsewhere)
- Tuesday, 3/13 : Review of Riemann sum (from Calc I), Double Riemann Sum, its geometric interpretation, and the definition of double integral of a 2-var function on a rectangle, Properties of double integral, Midpoint rule, Examples, Iterated integrals. (From Sections 16.1 and 16.2)
- Thursday, 3/15 : Fubini's Theorem - application to evaluation of double integral, Setting up a volume integral, Fubini's theorem for general (non-rectangular) domains, Double integrals over non-rectangular regions of type I and types II, Interplay between set up of type I and type II double integrals, Handout for identification of type of domain. (From Sections 16.2 and 16.3)
- Friday, 3/16 : Quiz#8 and discussion of its solutions, Examples for setting up and solving double integrals over type I and type II domains, Interplay between set up of type I and type II double integrals. (From Section 16.3)
- Tuesday, 3/27 : [Guest Lecture by Pat McCray] An example for volume using double integral, How (and when) to change the order of integration in a double integral, Some properties of double integrals including finding Area of a domain, and bounding a double integral using the area of the domain. (From Section 16.3)
- Thursday, 3/29 : How (and when) to change the order of integration in a double integral, Double integrals using polar coordinates, Describing the domains in terms of polar coordinates and solving double integrals over polar regions, (Using trigonometric integrals from Calculus II). (From Section 16.4)
- Friday, 3/30 : Quiz#9 and discussion of its solutions, Triple integral as a Triple Riemann Sum, Fubini's Theorem for Triple Integrals over a rectangular Box, Triple Integrals over general solids and their reduction to a double integral, Triple Integrals over solid regions of Type I and the iterated integrals for it. (From Section 16.6)
- Tuesday, 4/3 : Triple Integrals over solid regions of Type I, Type II, Type III, and the iterated integral for each, An example of how to describe a solid as Type I/ II/ III, Expressing the solid as a particular type could simplify the integral, Volume as a Triple integral. (From Section 16.6)
- Thursday, 4/5 : Review for Exam#2; Definition of cylindrical coords and examples, Definition of spherical coords and examples. (From Sections 16.7 and 16.8)
- Friday, 4/6 : Midterm Exam #2.
- Tuesday, 4/10 : Triple integral using Cylindrical coordinates and using Spherical coordinates - examples. (From Sections 16.7 and 16.8)
- Thursday, 4/12 : Midterm Exam #2: score distribution and the solutions. Change of variable in integrals, Image of S under a transformation. (From Section 16.9)
- Friday, 4/13 : Quiz#10 and discussion of its solutions, Change of variable in integrals, Image of S under a transformation, Jacobian of a transformation, How to find image of a transformation, find the Jacobian and simplify a double integral using a change of variables, examples. (From Section 16.9)
- Tuesday, 4/17 : Finding the appropriate Jacobian and simplifying a double integral using a change of variables, Jacobian and change of variable for triple integrals, examples, Line integral with respect to arc length- definition and change of integral to a 1-variable integral w.r.t. the underlying parameter, Line integral over piece-wise smooth curve. (From Section 17.2)
- Thursday, 4/19 : Mass and center of mass of a wire, Line integral w.r.t. variables x or y, Line integral of a 3-variable function over a space curve, Line integral that gives the length of the curve, Line integral over a space curve w.r.t. variables x or y or z, Line integral over a space curve w.r.t. variables x or y or z, Vector fields in the plane and in the space, gradient vector field, Work done by moving a particle over a curve in terms of a line integral, Line integral of a vector field on a curve, Example for work as a line integral; Fundamental Theorem of Line Integrals and conservative vector fields. (From Sections 17.2 and 17.3)
- Friday, 4/20 : Quiz#11 and discussion of its solutions. Fundamental Theorem of Line Integrals and conservative vector fields, Line integral independent of path and its relation to conservative fields over open and connected regions, conservative vector fields and their relation to a first order partial derivative condition for component functions, How to find the f such that F is the gradient of F when F is conservative vector field. (From Sections 17.3, 17.4)
- Tuesday, 4/24 : Examples for application of FTOLI, Positive orientation, Greens Theorem and how to apply it to evaluation of line integrals of non-conservative vector fields, finding area using line integral, Surface area of a graph of a function. (From Sections 17.3, 17.4, and 17.6)
- Thursday, 4/26 : Parametric surfaces, Finding parametrization of surfaces like cylinders(or parts of), spheres(or parts of), graph of a function, etc., Computing surface area of parametric surface, Definition of a surface Integral, Surface integral when the surface is a graph of a function, when surface has a parametric representation, Surface integral of vector field over an oriented surface using the unit normal vector, Stokes Theorem and Curl of a vector Field. (From Sections 17.5, 17.6, 17.8)
- Friday, 4/27 : Review for Final Exam.
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