Math 251 - Multivariable Calculus
Special office hours for the final
UPDATED!
MON through
THU, 9:00-10:00am and 2:00-3:00pm
I'm not available on Friday.
Subject to change! Always check here first.
Review and problems involving integrals
Click here to see the document that was distributed during recitation on 4/24, containing a review of all types of integrals and problems about them.
Important dates
First midterm: March 15, the Friday right before Spring break.
Second midterm: April 26, Friday
Final: May 14, Tuesday (8:00-11:00am at TIL-232)
NOTE THE TIME!
Maple Lab: Due
April 3.
MAPLE LAB
Due on April 3rd, Wednesday, as a printed copy during recitation
Click here to see the instructions.
You will need to use the data points
assigned to you, which you can find
here.
You are not expected to purchase Maple to complete this assignment. You can use it in the computer labs at Rutgers. But please don't leave this for the last day before the due date, since you may encounter problems using the software.
Recitations
- Section 5: WED 8:40-10:00, BE-253 (Beck Hall, Livingston)
- Section 6: WED 10:20-11:40, BE-253 (Beck Hall, Livingston)
- Section 7: WED 12:00-1:20, BE-121 (Beck Hall, Livingston)
Office hours
- MON 1:40-3:00, Hill Center room 624
- FRI 12:00-1:20, Hill Center room 624
- Also by appointment if you can't make it to either of these. Just send me an e-mail within 24 hours of notice before the time when you'd like to come.
- Please do not drop by unanounced outside of the official hours or scheduled appointments. I may be in the office, but I'll be doing my own work.
Communication
I maintain a mailing list that uses your preferred e-mails for Rutgers communications. If you never set a preferred e-mail, this will be your Rutgers e-mail. Make sure to check it every week.
Material covered and suggested problems
Topics and their sections in the book:
Important! The list below is an attempt to organize the material covered in class by approximate topics and chronological order.
It may contain more than what was covered and may omit other things that were covered! The TA will always make an effort to keep it updated by communicating with the instructor. Happy studying, everybody!
- Vector algebra 12.1 12.2
- Dot product and angle 12.3
- Orthogonal vectors 12.3
- Magnitude of a vector 12.3
- Projection of a vector onto another 12.3
- Cross product 12.4
- Area of parallelogram, volume of parallelepiped 12.4
- Equation of a plane 12.5
- Partial derivatives 14.3
- Plane tangent to a graph 14.4
- Linearization and approximation 14.4
- Unconstrained optimization 14.7
- Gradient and directional derivative 14.5
- Constrained optimization (Lagrange multipliers) 14.8
- 2nd derivative test for classification of local extrema 14.7
- Chain rule for paths f(t) = <a(t),b(t),c(t)> 14.5
- General chain rule 14.6
- Integration in two variables 15.1
- Computing area of a 2D region through integration of a constant 15.1, 15.2
- Computing volume of a 3D region between the graphs of two 2-varible functions 15.2
- Integration in three variables 15.3
- Double integral in polar coordinates 15.4
- Triple integral in spherical coordinates 15.4
- Triple integral in cylindrical coordinates 15.4
- Jacobian for a change of variables in multiple integrals 15.6
- Divergence, curl 16.1
- Potential function for a vector field 16.1
- Line integral of vector functions 16.2
- Line integral of a vector function as related to physical work 16.2
- Conservative vector fields and the fundamental theorem of calculus 16.3
- Green's Theorem 17.1
- Parametrized surfaces 16.4
- Surface integrals of vectors functions (also called flux integrals) 16.5
- Stokes' Theorem 17.2
- Divergence Theorem 17.3
Suggested problems (and section in the book) by topic:
Important! The list below consists mainly of the first half of the exercises in each chapter. Do as many as you feel necessary from each group (note that the problems within each group are similar).
But on the exam you are not going to know the topic of a problem! In order to address this problem, notice that the book also has review problems at the end of each CHAPTER, not just SECTION. You should try those as well if you want to see whether you can identify the topic of a problem without me telling you!
- Compute the dot product 1-12 (12.3)
- Compute the angle 19-28 (12.3)
- Distributive properties of dot product 35-38 (12.3)
- Relation between dot product, angle, magnitude 33, 43-48 (12.3)
- Geometry, angles, dot product 49-50 (12.3)
- Projection onto vector 51-60, 65-70 (12.3)
- Compute the cross product 9-16 (12.4)
- Distributive properties of cross product 17-22 (12.4)
- Areas and volumes using cross product 36-47 (12.4)
- Equation of a plane given normal vector and a point in it 1-10, 13-16 (12.5)
- Equation of a plane given three points in it 17-20 (12.5)
- Equation of a plane given other descriptions 21-30 (12.5)
- Intersection of planes with other things 38-49 (12.5)
- Compute the partial derivative 2-5, 13-44 (14.3)
- Compute the (higher-order) partial derivative 57-72 (14.3)
- Find the tangent plane 1-10 (14.4)
- Find points where the tangent plane has a certain property 11,12 (14.4)
- Linearization and approximation of a given function 13-22 (14.4)
- Use linearization to approximate a numerical expression 23-28 (14.4)
- Optimization in a bounded domain 35-45 (14.7)
- Optimization in a bounded domain (where the domain is not clearly stated) 46-50, 52-55 (14.7)
- Lagrange multipliers 1-15 (one constraint), 41-47 (two constraints) (14.8)
- Lagrange multipliers (where the constraint is not clearly stated) 17-23 (14.8)
- Second derivative test 7-23 (14.7)
- Compute the gradient 5-8 (14.5)
- Chain Rule for vector function of one variable 9-20 (14.5)
- Compute the directional derivative 21-30 (14.5)
- Gradient as a normal vector to level curves/surfaces 41-48 (14.5)
- General Chain Rule 1-16 (14.6)
- Implicit differentiation and the Chain Rule 25-32 (14.6)
- Evaluate double integrals over rectangles 8-10, 19-42 (15.1)
- Evaluate double integrals by changing the order of integration 44-47 (15.1)
- Evaluate double integrals in general 2D domains 3-13, 17-24 (15.2)
- Rewrite an integral over a general 2D domain by changing the order of integration 25-36 (15.2)
- Evaluate double integrals over 2D domains that may require decomposition into 2 or more parts 39-43 (15.2)
- Find volume of region bounded between graphs of 2-variable functions 45-50 (15.2)
- Evaluate triple integrals over parallelepipeds 1-8 (15.3)
- Evaluate triple integrals over x-simple, y-simple or z-simple regions 9-19 (15.3)
- Find volume by triple integration 20-21, 31-36 (15.3)
- Evaluate double integrals using polar coordinates 1-20, 23-24 (15.4)
- Evaluate or re-express triple integrals using cylindrical coordinates 27-36 (15.4)
- Evaluate triple integrals using spherical coordinates 43-51 (15.4)
- Compute Jacobian determinants 11-18 (15.6)
- Evaluate double integrals by change of variables 29-32, 35-36 (15.6)
- Sketch vector fields / Recognize them by their sketches 1-20 (16.1)
- Finding potential functions 39-45 (16.1), 7-16 (16.3)
- Evaluate line integral (of vector function) 3-4, 7-8, 19-26 (16.2)
- Evaluate line integral of vector function denoted in Adx + Bdy + Cdz notation 27-32 (16.2)
- Line integral as physical work 51-58 (16.2)
- Evaluate line integrals of conservative vector fields 1-6 (16.3)
- Green's Theorem 3-10 (17.1)
- Making certain line integrals easier by using Green's Theorem 11-13 (17.1)
- Computing area by using Green's Theorem 14-17 (17.1)
- Parametrizing surfaces, computing their area, identifying the normal vector 1-4, 30 (16.4)
- Finding tangent plane to a surface 7-10 (16.4)
- Evaluate surface integral (of vector function) 5-17, 20-21 (16.5)
- Surface integral of vector function as a flux 25-28 (16.5)
- Stokes' Theorem 1-4 (17.2)
- Stokes' Theorem to compute a flux integral 5-10 (17.2)
- Stokes' Theorem to compute a line integral 11-14 (17.2)
- Divergence Theorem 1-4 (17.3)
- Divergence Theorem to compute a flux integral 5-16 (17.3)
Quizzes
- 20 minutes in the end of recitation every Wednesday.
- Covers the material taught up to and including the previous Friday in class.
- Material is cumulative (things learned at the beginning of the semester may be asked, directly or indirectly because they're needed for the newer stuff), but emphasis is always on the most recent stuff.
- Calculators and books/notes are never allowed.
- The 2 lowest scores are dropped.
- There are no make-ups ever, for whatever reason. The "drop 2 lowest" policy is intended to cover your emergencies and excused absenses. In the rare case a student has a valid reason to miss 3 or more quizzes, then he or she should talk to the TA to determine a good way to make them up.
Solutions: