Homework will be completed and assigned on WebAssign. Please enroll in WebAssign via Brightspace.
Week 15
Reading: Section 12.8
Class on Monday, December 15 will be treated as office hours. Please come with questions from your studying for the final exam.
HW15 will not be graded. It is for you to practice change of variables problems before the final exam.
Final Exam Information:
The final exam is on Wednesday, December 17 1:45–3:45pm in Kiely 242 (our standard room).
The final exam is cumulative but about 50% of the exam will cover material from after Exam 2 (so multiple integrals).
During the semester, we covered the following sections from the textbook: 9.1–4, 10.1–9, 11.1–8, 12.1–8.
You may bring two sheets of notes to the exam.
Your notes may contain definitions, formulas, and statements of theorems, not worked out problems.
Monday, December 8 (Day 26): Derived the change of variables formula for double integrals. Went over several examples of using the formula.
Wednesday, December 10 (Day 27): Worked out another examples of change of variables. Introduced the change of variables formula for triple integrals and worked through two examples.
Week 14
Reading: Section 12.4, 12.5, 12.6, 12.7, 12.8
Monday, December 1 (Day 24): Began with an example of computing a triple integral. Discussed several applications of double and triple integrals, including volume, mass, center of mass, probabilities, and expected value. Introduced cylindrical coordinates and derived the formula for triple integrals in cylindrical coordinates.
Wednesday, December 3 (Day 25): Introduced spherical coordinates. Derived the formula for the volume element in spherical coordinates, allowing us to compute triple integrals with respect to spherical coordinates. Introduced the notion of a transformation between two regions in the plane.
Week 13
Reading: Section 12.3
Monday, November 24 (Day 22): Derived the formula for double integrals in polar coordinates.
Wednesday, November 26 (Day 23): Worked through an example of computing a double integral in polar coordinates over a polar domain. Introduced triple integrals over a box and a general solid in R^3. Worked through an examples.
Week 12
Reading: Section 12.2
Monday, November 17 (Day 20): Introduced double integrals over a general domain and worked through examples.
Monday, November 19 (Day 21): Exam 2
Week 11
Reading: Sections 11.8, 12.1
Exam 2 is on Wednesday, November 19. The exam will cover Chapter 11 in the textbook. This corresponds to homework assignments 7 through 11.
Monday, November 10 (Day 18): Went over two more extremal value problems. Introduced the method of Lagrange multipliers.
Wednesday, November 12 (Day 19): Introduced double integrals over a rectangle. Discussed Fubini's theorem and went over examples of computing double integrals.
Week 10
Reading: Sections 11.7
Monday, Nov. 3 (Day 16): Introduced the gradient vector and showed that the maximum value of the directional derivative is equal to the magnitude of the gradient and is in the direction of the gradient. Derived the orthogonality of the gradient vector to level curves/surfaces. Defined critical points and showed that local maximums and local minimums occur at critical points.
Wednesday, Nov. 5 (Day 17): Introduced the second derivative test and the Extreme Value Theorem. Worked through examples.
Week 9
Reading: Sections 11.5 and 11.6
Monday, Oct. 27 (Day 14): Recalled the definition of differentiability for a two-variable function. Went through an example of a function all whose partial derivatives exist but fails to be differentiable. Stated a theorem guaranteeing differentiability in terms of the partial derivatives. Introduced and proved the chain rule in higher dimensions. Reviewed implicit differentiation in a single variable.
Wednesday, Oct. 29 (Day 15): Discussed implicit differentiation for equations in three variables and the associated implicit function theorem. Introduced directional derivatives.
Week 8
Reading: Sections 11.3 and 11.4
We have class on Friday, Oct. 24, as CUNY is following a Monday schedule.
Wednesday, Oct. 22 (Day 12): Discussed basic properties of limits of functions in two (or more) variables. Defined continuity for functions of two (or more) variables. Introduced polynomials and rational functions in several variables. Defined partial derivatives and interpreted as slopes of tangent lines to curves obtained as intersections with the graph surface and the planes obtained from holding values of either x or y constant.
Friday, Oct. 24 (Day 13): Discussed second partial derivates and stated Clairut's theorem. Introduced the notion of a tangent plane to the graph of a two-variable function and derived its equation. Finished with defining differentiability for a two-variable function.
Week 7
Reading: Sections 11.1 and 11.2
Exam 1 is on Wednesday, October 15. See Week 6 notes for details.
No class Monday, October 20, but we have a make-up class on Friday, October 24.
A new homework will be posted on Wednesday, Oct. 15, with a due date the following week.
Tuesday, Oct. 14 (Day 10): Introduced level curves and used them to sketch the graphs of several functions. Introduced limits for functions of two variables. Worked through an example of showing how the limit fails to exist.
Wednesday, Oct. 15 (Day 11): Exam 1
Week 6
Reading: Sections 10.9, 10.6, and 11.1
Exam 1 is on Wednesday, October 15. You may bring one sheet of notes—containing only definitions, formulas, and statements of theorems—to the exam.
Exam 1 will cover the following sections of the book: 9.1 through 9.4 and Chapter 10 (with the exception of Section 10.6 and of the subsections on work and torque). This corresponds to the first six weeks of classes and the first six homework assignments on WebAssign.
There is no class on Monday, Oct. 13, but we do have class on Tuesday, Oct. 14 at the usual time and place.
Monday, Oct. 6 (Day 8): Computed the curvature of a circle. Derived a formula for curvature depending only on a vector function and its derivatives (as opposed to the derivative of the unit tangent vector). Defined the normal and binormal vectors, the normal and osculating planes, and osculating circles. Discussed motion in space: position, velocity, and acceleration vectors. Went over the classical problem of projectile motion. Discussed the component formula for acceleration, noting accelerating has no component in the binormal direction. Defined a surface and introduced cylinders.
Wednesday, Oct. 8 (Day 9): Introduced quadratic surfaces and the notion of sections (or traces) of surfaces (ie, the intersection of a surface with a plane). Worked through several examples of equations of quadratic surfaces, investigating their various sections and using this information to visualize the surface in three-dimensional space. Introduced the notion of a function in two variables and the graphs of such functions.
Week 5
Reading: Sections 10.7 and 10.8
There is a new WebAssing due next week covering the material from Monday's class.
As mentioned in class, you should know how to derive properties 3, 4, and 6 from Theorem 5 in section 10.7.
Monday, Sept. 29 (Day 7): Started by deriving a formula for the distance of a point from a plane. Introduced vector functions and space curves. Discussed derivates and integrals of vector fuctions as well as differentiation rules. Began our discussion of curvature, giving the definition and proving that a line has zero curvature.
Week 4
Reading: Sections 10.3, 10.4, 10.5
No office hours on Wednesday.
Monday, Sept. 15 (Day 5): Defined vector and scalar projections of one vector onto another. Introduced the cross product. Gave determinant version of cross product, related magnitude of cross product to the sine of the angle between the vectors and to the area of the parallelogram spanned by the vectors. Gave basic properties of the cross product.
Wednesday, Sept. 17 (Day 6): Derived equations of lines and planes in 3-space.
Week 3
Reading: Sections 9.3, 9.4, 10.1, 10.2, 10.3
I extended the deadline for HW1 and HW2 until Saturday, as some students had difficulty getting WebAssign up and running.
Monday, Sept. 8 (Day 3): Derived formula for area under polar curve and length of polar curve. Introduced 3-dimensional coordinate system. Introduced vectors.
Wednesday, Sept. 10 (Day 4): Defined vector addition and scalar multiplication. Introduced components of vectors and how to compute addition of vectors, scalar multiplication, and the magnitude of a vector in components. Defined the dot product and used the law of cosines to derive the relationship between the dot product and the angle between two vectors.
Week 2
Reading: Section 9.2, 9.3, and 9.4
Office hours on Wednesday, September 3 will be moved to 3:30–4:30pm (due to a department meeting).
There are now two assignments available on WebAssign; both are due next Wednesday, Sept. 10.
Wednesday, Sept. 3 (Day2): Computed the length of one pass of the cycloid. Introduced polar coordinates, polar equations, and polar curves. Derived the formula for the slope of the tangent line to a polar curve when the radius is a function of the angle.
Week 1
Reading: Sections 9.1 and 9.2
There will be a WebAssign posted for the first day of class. It's due date will be in two weeks to give people time to sign up for WebAssign.
Wednesday, Aug. 27 (Day 1): Introduced parametric curves. Computed tangent lines to parametric curves. Derived formula for arc length of a parametric curve.