Math 251 : Multivariable and Vector Calculus
Instructor: Hemanshu
Kaul
Office: 234B, Engineering 1
Phone: (312) 567-3128
E-mail: kaul [at]
math.iit.edu
Time: 11:25am, Monday-Wednesday-Friday.
Place: 108, Perlstein Hall.
Office Hours: 3:30pm-4:30pm Monday and Wednesday, walk-ins, and by appointment. Emailed questions are also encouraged.
Tutoring Service: Mathematics tutoring at the Academic Resource Center.
Online Problem Practice: Calculus III book at COW (Calculus on Web).
|Course Information|
|Advice|
|Announcements|
|Examinations|
|Homework|
|Class Log|
|Links|
Course Information:
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? Basics, More Details.
How to study and learn math.
Understanding Mathematics - a study guide
Class Announcements:
- Wednesday, 12/5 : All HW solutions have been distributed in class.
- Wednesday, 11/14 : All HW solutions up to HW#12 have been distributed in class.
- Friday, 10/12 : Mid-term Exam #2 has been announced with CORRECT date. Look below in the appropriate section.
- Tuesday, 9/18 : Mid-term Exam #1 has been announced. Look below in the appropriate section.
- Friday, 9/7 : HW#1 solutions were distributed in class. Future HW solutions will also be distributed in class.
- Wednesday, 8/29 : A forum for homework discussion is now available on the discussion board in the IIT Blackboard website for this course. You can access blackboard through myIIT or directly at IIT blackboard.
- Friday, 8/24 : Every student has to send me an email regarding their aim for this course.
- Friday, 8/24 : Check this webpage regularly for homework assignments, announcements, etc.
Examinations:
- Exam # 1 : October 5th, Friday. Topics: Everything covered in class up to and including September 26th, Wednesday.
- Exam # 2 : November 19th, Monday. Topics: Everything covered in class from and including September 28th, Friday up to and including November 7th, Wednesday.
- Final Exam : Monday, December 10, 2pm to 4pm. Topics: Everything done during the semester.
Homework Assignments:
The assignments below indicate the problems that you should thoroughly understand in order to do well in class. If you need to practice more - the book provides many more problems (Remember that the Student's Solution Manual has solutions to all odd numbered problems). Turn in only the blue colored problems below.
- Homework #1 : Due Friday, 8/31. HW#1 solutions distributed in class on 9/7, Friday.
- Friday, 8/24 : Section 13.1: 3, 6, 8, 10acf, 12, 16, 17, 22, 24, 29, 30, 31, 32, 35, 36.
- Monday, 8/27 : Section 13.2: 5, 8, 9, 12, 13, 16, 17, 20, 22, 35.
- Homework #2 : Due Friday, 9/7. HW#2 solutions distributed in class on 9/14, Friday.
- Wednesday, 8/29 : Section 13.2: 25, 26. Section 13.3: 2, 6, 8, 9, 14, 21, 24bc, 26, 39, 40, 57.
- Friday, 8/31 : Section 13.3: 41, 47. Section 13.4: 3, 5, 7, 8, 11, 12, 18, 24, 28.
- Homework #3 : Due Friday, 9/14. HW#3 solutions distributed in class on 9/26, Wednesday.
- Wednesday, 9/5 : Section 13.4: 35, 38, 39a, 42. Section 13.5: 4, 11, 14, 16.
- Friday, 9/7 : Section 13.5: 20, 21, 26, 29, 33, 36, 38, 44, 48, 49.
- Homework #4 : Due Wednesday, 9/19. HW#4 solutions distributed in class on 9/28, Friday.
- Monday, 9/10 : Section 13.5: 51, 54, 61, 68, 71, 72.
- Wednesday, 9/12 : Section 13.6: 11, 12, 15, 16, 20, 21-28, 30, 33, 34, 41, 43, 44. Section 13.7: 3, 10, 11, 15, 19, 22, 23, 26, 27, 28, submit any one of 31-36.
- Homework #5 : Due Wednesday, 9/26. HW#5 solutions distributed in class on 10/1, Monday.
- Friday, 9/14 : Section 13.7: 38, 39, 42, 44, 45, 53, 56, 59, 60. Section 14.1: 3, 4, 5, 6, 9, 10, 12, 19-24, 34.
- Monday, 9/17 : Section 14.2: 9, 12, 13, 14, 17, 20, 23, 26, 35, 36, 37.
- Homework #6 : Due Monday, 10/1. Last HW before midterm exam #1. HW#6 solutions distributed in class on 10/1, Monday.
- Wednesday, 9/19 : Section 14.3: 1, 4, 5, 6, 10, 11, 12, 14(you don't have to find the unit normal vector), 15, 18, 19, 23, 24.
- Friday, 9/21 : Section 15.1: 7, 8, 11, 15, 18, 20, 22, 25, 26 (describe the graph), 38 (describe the level curves), 41, 42, 60, 61.
- Monday, 9/24 : Section 15.2: 7, 8, 11, 12, 14, 15, 16, 37, 38 (Hint: Use L'Hospital's rule), 39.
- Homework # 7: Due Friday, 10/12.HW#7 solutions distributed in class on 10/7, Wednesday.
- Wednesday, 9/26 : Section 15.2: 19, 20, 23, 24, 29, 30, 32, 33.
- Friday, 9/28 : Section 15.3: 15, 16, 18, 19, 22, 24, 29, 30, 41, 42, 47, 49, 54, 59, 60.
- Monday, 10/1 : Section 15.4: 4, 5, 13, 14, 15, 19.
- Homework # 8: Due Wednesday, 10/17.HW#8 solutions distributed in class on 10/29, Monday.
- Monday, 10/8 : Section 15.4: 23, 25, 26, 27, 28, 29, 31, 32.
- Wednesday, 10/10 : Section 15.5: 3, 4, 5, 7, 9, 10, 12, 14, 17, 19, 22, 23, 24, 25, 28, 29, 31, 34.
- Homework # 9: Due Friday, 10/26. This is a longer HW than usual, start working on it right away (you have more than a week to complete it).HW#9 solutions distributed in class on 11/7, Wednesday.
- Friday, 10/12 : Section 15.6: 5, 6, 7, 8, 9, 13, 14, 15, 16, 23, 24, 25, 31, 40, 41, 52.
- Monday, 10/15 : Section 15.7: 7, 9, 10, 17, 18, 28, 29, 31, 32, 37, 40, 42, 45.
- Wednesday, 10/17 : Section 15.8: 5, 8, 9, 13, 15, 16, 18, 23, 24, 39.
- Homework # 10: Due Friday, 11/2.HW#10 solutions distributed in class on 11/12, Monday.
- Wednesday, 10/24 : Section 16.1: 1, 4, 11, 12, 13.
- Friday, 10/26 : Section 16.2: 1, 5, 8, 11, 12, 13, 14, 19, 20, 23, 26, 28, 29.
- Homework # 11: Due Wednesday, 11/7. HW#11 solutions distributed in class on 11/14, Wednesday.
- Monday, 10/29 : Section 16.3: 7, 11, 13, 14, 15, 18, 19, 20, 23, 24, 28, 44, 45, 48, 50.
- Wednesday, 10/31 : Section 16.4: 9, 11, 14, 16, 18, 19, 22, 23, 24, 27, 29.
- Homework # 12: Due Wednesday, 11/14. Last HW before midterm exam #2. HW#12 solutions distributed in class on 11/14, Wednesday.
- Friday, 11/2 : Section 16.6: 2, 3, 6, 7, 9, 10, 12.
- Monday, 11/5 : Section 16.7: 8, 11, 12, 14, 15, 16, 19, 20.
- Wednesday, 11/7 : Section 16.8: 8, 10, 12, 13a, 18, 19, 20, 23, 33, 34, 35, 36.
- Homework # 13: Due Friday, 11/30. HW#13 solutions distributed in class on 12/3, Monday.
- Friday, 11/9 : Section 16.9: 3, 4, 6, 8, 9, 10, 12, 14, 20, 22, 23.
- Monday, 11/12 : Section 17.2: 3, 4, 5, 8, 10, 14, 15, 19, 20, 22, 32.
- Wednesday, 11/14 : Section 17.1: 11-14, 29-32.
- Wednesday, 11/21 : Section 17.3: 4, 5, 6, 8, 13, 16, 17, 18, 22.
- Homework # 14: Due Wednesday, 12/5. Last HW to be submitted before Finals. HW#14 solutions distributed in class on 12/5, Wednesday.
- Monday, 11/26 : Section 17.4: 2, 4, 7, 9, 10, 12, 13, 14.
- Wednesday, 11/28 : Section 17.6: 17, 18, 19, 21, 22, 37, 38, 40, 41, 44.
- Friday, 11/30 : Section 17.7: 5, 8, 9, 10, 14, 15, 20, 21, 24.
Class Log:
- Friday, 8/24 : Introductions, 3-D geometry, coordinate axes & planes, notation and distance formula for points in the space, equations of planes parallel one of the coordinate planes, examples for description of surfaces and bodies in the space, equation of sphere; Handout (with solutions) for surfaces. (From Section 13.1)
- Monday, 8/27 : Course requirements and aim; vectors - algebraic and geometric perspectives of addition, scalar multiplication and difference, length of a vector, properties of vectors, unit vectors, i,j,k vectors, unit vectors; Handout for vector addition. (From Section 13.2)
- Wednesday, 8/29 : 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, No cancelation law for dot product, scalar and vector projections; Handout for practice. (From Sections 13.2 and 13.3)
- Friday, 8/31 : Discussion of the previous Handout; 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 and its relation to area of a parallelogram, condition for two vectors to be parallel, examples. (From Sections 13.3, 13.4)
- Wednesday, 9/5 : cross product of basis vectors, non-properties and properties of cross product, torque as a cross product; Handout for practice and discussion of problems; 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, 9/7 : Vector, parametric, and symmetric equations of a line in 3-D space and the relation between the three definitions, 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; 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, angle between two planes, Equation of line of intersection of two planes. (From Section 13.5)
- Monday, 9/10 : Equation of line of intersection of two planes, Distance from a point to a plane, distance between two parallel or skew lines; Cylinder in 3-D, Quadric surfaces, examples - parabolic cylinder, circular cylinder, ellipsoid, hyperboloid, etc., Handout for practice. (From Sections 13.5 and 13.6)
- Wednesday, 9/12 : Examples and discussion of Handout for section 13.6; Cylindrical coordinates - their geometric meaning and their relation to polar coordinates in 2-D, equations relating rectangular coords and cylindrical coords, Spherical coordinates - their geometric meaning, equations relating rectangular coords and spherical coords, some standard examples - r=c, theta=c, rho=c, theta=c, phi=c; Handout for practice. (From Sections 13.6 and 13.7)
- Friday, 9/14 : Examples for Spherical coordinates and underlying trigonometry; Vector-valued functions- domain, limit, and continuity, Relation between vector functions and parametric equation of a curve in 3-D space, examples; Handout for practice. (From Sections 13.7 and 14.1)
- Monday, 9/17 : Discussion of previous handout, Derivative of a vector function and its relation to the component functions, Tangent vector and Tangent line - their relation, unit tangent vector, Examples, Differentiation rules for vector functions, definite and indefinite integrals of vector functions. (From Sections 14.1 and 14.2)
- Wednesday, 9/19 : 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, Curvature - definition in terms of arc length, alternate formula in terms of the original parameter, another formula using second-order derivative and cross product, examples, curvature for plane curves. (From Section 14.3)
- Friday, 9/21 : Curvature for plane curves; functions of two variables - domain, range, graphs (relation to material from 13.5 and 13.6), level curves. (From Sections 14.3 and 15.1)
- Monday, 9/24 : 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, Methods for existence of limit - substitution in terms of distanc3, substitution in terms of polar coordinates. (From Sections 15.1 and 15.2)
- Wednesday, 9/26 : Limit laws and squeeze theorem for 2-variable functions, 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. (From Sections 15.2 and 15.3)
- Friday, 9/28 : Geometric interpretation of partial derivative, higher order partial derivatives, Clairut's Theorem and its application, an example of a PDE. (From Section 15.3)
- Monday, 10/1 : Tangent Plane to a surface given by a 2-variable function and its relation to the partial derivatives, 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, corresponding concepts for 3 or more variable functions. (From Section 15.4)
- Wednesday, 10/3 : Review session for Midterm Exam #1.
- Friday, 10/5 : Midterm Exam #1.
- Monday, 10/8 : 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 for multivariable functions. (From Section 15.4 and 15.5)
- Wednesday, 10/10 : Examples for General chain rule, Formulas for implicit differentiation; Distribution and discussion of Mid-term Exam #1 with solutions. (From Section 15.5)
- Friday, 10/12 : Directional derivatives - definition and formula using gradient, Gradient of a functions, generalization to 3-variables, Method of steepest ascent/ descent using gradient, Formula for a tangent plane to a level surface of a 3-variable function. (From Section 15.6)
- Monday, 10/15 : Equation of the Normal line to a surface at a given point, Formula for a tangent plane to a level curve of a 2-variable function, local maximum and minimum of 2-variable functions, critical points, Second derivative test. (From Sections 15.6 and 15.7)
- Wednesday, 10/17 : Extreme value theorem and extreme points on the boundary, 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. (From Sections 15.7 and 15.8)
- Monday, 10/22 : Class Canceled.
- Wednesday, 10/24 : Example for Method of Lagrange Multipliers for two constraints, Review of Riemann sum, 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. (From Sections 15.8 and 16.1)
- Friday, 10/26 : Discussion of Problem 15.7.18; Iterated integrals, Fubini's Theorem - application to evaluation of double integral, Setting up a volume integral, Fubini's theorem for general (non-rectangular) domains. (From Sections 16.2 and 16.3)
- Monday, 10/29 : Double integrals over non-rectangular regions of type I and types II, Interplay between set up of type I and type II double integrals, Identification of the domain and the integrand for volume integrals, 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)
- Wednesday, 10/31 : Double integrals using polar coordinates, Describing the domains in terms of polar coordinates, Using trigonometric integrals from Calculus II, Surface area of a graph of a function over a domain. (From Sections 16.4 and 16.6)
- Friday, 11/2 : Using polar coordinates to fins surface area, 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. (From Sections 16.6 and 16.7)
- Monday, 11/5 : Triple Integrals over solid regions of Type II & III, An example of how to describe a solid as Type I/ II/ III, Expressing the solid as a particular type could simply the integral, Volume as a Triple integral, Triple integral using Cylindrical coordinates. (From Sections 16.7 and 16.8)
- Wednesday, 11/7 : Triple integral using Spherical coordinates, Change of variable in double integrals, Jacobian of a transformation, How to find the image of a transformation. (From Sections 16.8 and 16.9)
- Friday, 11/9 : How to find image of a transformation, find the Jacobian and simplify a double integral using a change of variables, Jacobian and change of variable for triple integrals, examples. (From Section 16.9)
- Monday, 11/12 : 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, Mass and center of mass of a wire, Line integral w.r.t. variables x or y. (From Section 17.2)
- Wednesday, 11/14 : 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, 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 and its it relation to line integral w.r.t. x, y and z. (From Sections 17.2 and 17.1)
- Friday, 11/16 : Review session for Midterm Exam #2.
- Monday, 11/19 : Midterm Exam #2.
- Wednesday, 11/21 : Another example for Work as a line integral; 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 Section 17.3)
- Monday, 11/26 : More examples for application of FTOLI, Positive orientation, Greens Theorem and how to apply it to evaluation of line integrals of non-conservative vector fields, Green's Theorem applied to finding area. (From Section 17.4)
- Wednesday, 11/28 : More examples for Green's Theorem, Parametric surfaces, Identifying 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. (From Sections 17.4, 17.6)
- Friday, 11/30 : More examples for Parametric representations and surface area, 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, the formula for unit normal vector of a parametric surface. (From Sections 17.6, 17.7)
- Monday, 12/3 : Examples for practicing surface integration when the surface is a graph of a function, when surface has a parametric representation, and surface integration of vector field over an oriented surface using the unit normal vector. (From Section 17.7)
- Wednesday, 12/5 : Stokes Theorem and Curl of a vector Field, Gauss' Divergence Theorem and Div of a vector field. (From Sections 17.5, 17.8 and 17.9)
- Friday, 12/7 : Review session for Final Exam.
Links for Additional Information:
HOME