See the schedule of free tutoring which begins tonight. Note that my own office hours have been modified for this week only.

My office hours have been changed from today until Thursday (4-5:30pm). See the schedule of free tutoring.

We discussed functions of several variables, along with domains, graphs of z = f(x,y), level curves, etc.

We then compared the techniques for finding limits of functions from R to R versus limits of functions from R² to R. If the function is continuous at a given point, we can just plug the value into the function to obtain the limit. Luckily many of the functions we deal with are continuous on their natural domains.

In calculus I, we learned that for a limit to exist, the limit from the left and from the right had to be equivalent. Unfortunately for z = f(x,y), there are many paths to a given point (not just from the left and right). Also l'Hospital's Rule cannot be used for functions from R² to R. We saw that (x^2 - y^2) / (x^2 + y^2) has different limiting values when (x,y) approaches (0,0) along the x-axis versus approaching (0,0) along the y-axis. We then investigated the limit of (x*y^2) / (x^2 + y^4) as (x,y) approaches (0,0) along y = m*x versus approaching along x = y^2. We'll look mare carefully at examples of limits next time along with the formal definition of limits.

Check the available tutoring hours. No appointment is necessary.

The cover page on your test will include a seating chart. When you pick up your test on Wednesday, one of these seat numbers will be circled and that will be your assigned seat for the first test. Try to arrive early to obtain your assigned seat. Unless you happen to sit at the end of a row, you will be required to stay for the full 50 minute testing period. The TAs will not answer any questions during the test.

Be sure to go to both discussion sections this week.

In lecture we looked at the example f(x,y) = x² y². We found the critical points to be (a, 0) and (0, b) where a and b are any real numbers. For these values we saw that the discriminant D = 0. For this reason the 2nd derivative test is inconclusive. However we saw in another way that the absolute minimum value of 0 is obtained at each of these critical points.

We discussed the Extreme Value Theorem and where to look for absolute maxima and absolute minima. In particular we found the absolute max/min of f(x, y) = x² + y² − 2x + 2y + 5 on the set of (x,y) pairs for which x² + y² ≤ 4. For the boundary we used the parameterization x = 2 cos(t), y = 2 sin(t), 0 ≤ t ≤ 2π. The book's solutions show another approach for a similar problem.

Lagrange Multipliers can be used to minimize or maximize a function f(x,y,z) subject to a constraint g(x,y,z) = c. Our particular example used fewer variables to help us visualize why this technique works. We saw how to find the point closest to the origin on the hyperbola xy = 3. This amounted to minimizing the function f(x,y) = x² + y² subject to the constraint xy = 3. Although this problem can be solved without using Lagrange Multipliers, it is helpful to know this technique for problems which cannot be solved easily in other ways.

Next time we will look at more complicated examples of Lagrange Multipliers. I may also introduce section 15.1.

We sought to minimize f(x,y,z) = 2x² + y² + 3z² under the constraint 2x − 3y = 49 + 4z. We needed to rearrange the constraint equation so that we had only a constant on the right hand side. This gave 2x − 3y − 4z = 49 and informed us to let g(x,y,z) = 2x − 3y − 4z. Using the technique of Lagrange Multipliers we set ∇ f = λ ∇ g to arrive at the equations 4x = 2λ, 2y = −3λ and 6z = −4λ. Together with the constraint equation 2x − 3y − 4z = 49, we solved the system of equations to obtain (x,y,z) = (3, −9, −4). Graphically we should see why there is no maximum, so this point must give us the minimum value f(3, −9, −4) = 147.

When we wish to find the extreme values of f(x,y,z) subject to the two constraint equations g(x,y,z) = C₁ and h(x,y,z) = C₂, the technique of Lagrange Multipliers generalizes to solving the system of equations obtained from f = λ₁ ∇ g + λ₂ ∇ h along with the two constraint equations.

I introduced volumes over a rectangular region as a limit of double Riemann sums. I defined double integrals using this limit and we used it to determine certain volumes. If f is continuous over a rectangular region, then Fubini's theorem tells us we will obtain the same result regardless of whether we integrate with respect to x first then y or with respect to y first then x. For the homework problems you should always solve the double integrals both ways if possible. It is only after attempting this for each problem that you will learn to anticipate which approach will be easiest.

Check the available tutoring hours. No appointment is necessary.

The cover page on your test will include a seating chart. When you pick up your test on Wednesday, one of these seat numbers will be circled and that will be your assigned seat for this test. Try to arrive early to obtain your assigned seat. Unless you happen to sit at the end of a row, you will be required to stay for the full 50 minute testing period. The TAs will not answer any questions during the test.

Be sure to go to both discussion sections this week.

We discussed change of variables in the following contexts.

• We solved a simple integral using u-substitution as seen in Calculus I.
• We solved #28 in 15.7 by switching from rectangular to cylindrical coordinates.
• We solved #39 in 15.8 by switching from rectangular to spherical coordinates.
• We found the area of the ellipse x²/a² + y²/b² = 1 by substituting u = x/a and v = y/b.
• We saw that the change in variables u = 3x − 2y and v = x + y resulted in dxdy = (1/5)dudv so that a double integral with respect to x and y can be converted to a double integral in terms of u and v.
• I quickly defined the Jacobian and showed how it is used to transform dxdy into dudv in order to change the variables of integration.

I discussed the Jacobian in greater detail today and used it in the following problems.

• We saw that x = r cos(θ) and y = r sin(θ) leads to dy dx = r dr dθ as expected when converting from rectangular to polar coordinates.
• We saw that x = ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ) and z = ρ cos(φ) leads to dz dy dx = ρ² sin(φ) dρ dθ dφ as expected when converting from rectangular to spherical coordinates.
• To evaluate the double integral of sqrt( (x − y) / (x + y + 1) ) over the square with vertices (0, 0), (1, −1), (2, 0), and (1, 1), we used the change of variables u = x − y and v = x + y. I believe that I mistakenly used the partials of u and v with respect to x and y instead of the partials of x and y with respect to u and v to compute the Jacobian. This resulted in a Jacobian of 2 instead of the correct Jacobian of 1/2. I'll post a corrected solution here this weekend.
• We began the set-up of #15 in section 15.9 noting that the region is not transformed into a nice polygon.

I spent time in lecture discussing how to decide what change of variables to make. We mostly concentrated on simplifying the region of integration but also looked at simplifying the integrand.

For example, problem 15 from section 15.9 included boundaries given by the hyperbolas xy = 1 and xy = 3. By letting u = xy we now have u ranging from 1 to 3 and the region of integration is simplified. We also let v = y and did a careful analysis of how to solve this problem.

We also looked at problems 19, 20 and 21 from section 15.9 to see how we arrive at a reasonable substitution.

I introduced vector fields and mentioned their applications to wind patterns, fluid flow, force fields, electric/magnetic fields. We then looked at graphs for the following vector fields.

• F = 2i − j = ⟨2, −1⟩
• F = xi = ⟨x, 0⟩
• F = xi + yj = ⟨x, y⟩
• F = −yi + xj = ⟨−y, x⟩

In lecture we discussed smooth curves, parametrizations and line integrals. We saw that line integrals can be used to compute area, average value, mass, work, etc. We spent some time on deriving how to compute the work done by a force field in moving an object along a curve. For our examples we calculated the following.

• The average temperature along the border of a circular plate of radius 3 centered at the origin when the temperature is given by T(x,y) = 100(x² + 2y²).
• The work done by a force field F = −yi + xj = ⟨−y, x⟩ in moving an object along the curve C given parametrically by x = t, y = t², 0 ≤ t ≤ 1.

The plan for the rest of the semester is to cover sections 16.4 – 16.9 in order.

I began class by answering questions from earlier sections. We looked at the following equivalent conditions.

Let F = ⟨P, Q⟩ have continuous first partial derivatives in an open connected region R and let C be a piecewise smooth curve in R. The following conditions are equivalent.

• F is conservative. That is, F = ∇ f for some function f.
• The integral along C of F • dr is independent of path.
• The integral along C of F • dr = 0 for every closed curve C in R.

For our main example we let C₁ be the path from (0,0) to (2,0) along the top half of a circle of radius 1 centered at (1, 0). Given F = ⟨y³ + 1, 3xy² + 1⟩, we found the integral along C₁ of F • dr in three ways.

1. We parametrize C₁ using r(t) = ⟨1 − cos(t), sin(t)⟩ for 0 ≤ t ≤ π. Then dr / dt = ⟨sin(t), cos(t)⟩ and F = ⟨sin³(t) + 1, 3(1 − cos(t))sin²(t) + 1⟩. Now the integral along C₁ of F • dr = the integral from 0 to π of = F • (dr / dt) dt = the integral from 0 to π of ( sin⁴(t) + sin(t) + 3(1 − cos(t))sin²(t)cos(t) + cos(t) ) dt. This is very messy but the integral can be evaluated to give an answer of 2.
2. Using the techniques in section 16.3 we know that F is a conservative vector field (see theorem 6). We find a potential function f = xy³ + x + y. Now the integral along C₁ of F • dr = the integral along C₁ of ∇ f • dr = f(2,0) − f(0,0) = 2.
3. Since F is conservative, we know that the value of the line integral is independent of path. Thus we replace the semicircular path with the simpler straight line path C₂ from (0,0) to (2,0). The parametrization we use for C₂ is r(t) = ⟨t, 0⟩ for 0 ≤ t ≤ 2. Then dr / dt = ⟨1, 0⟩ and F = ⟨1, 1⟩. Now the integral along C₁ of F • dr = the integral along C₂ of F • dr = the integral from 0 to 2 of F • (dr / dt) dt = the integral from 0 to 2 of 1 dt = 2.

We looked at the statement and proof of Green's Theorem along with a number of examples.

We looked at an example with Green's Theorem when the curve was not positively oriented.

I proved the theorem given as problem #27 in section 16.3. Afterwards we looked at the definitions, physical interpretations and some theorems for curl F and div F.

The test on Wednesday will be on the following sections. I've included the section heading and the homework for each of these sections.

• Section 15.7 (Triple Integrals in Cylindrical Coordinates): #1, 3, 5, 9, 15, 16, 17, 21, 22.
• Section 15.8 (Triple Integrals in Spherical Coordinates): #1, 3, 7, 11, 12, 18, 23, 25, 27, 28, 34.
• Section 15.9 (Change of Variables in Multiple Integrals): #1, 2, 3, 7, 8, 12, 13, 15, 19, 20, 21, 22.
• Section 16.1 (Vector Fields): #1, 2, 6, 11, 15, 21, 23.
• Section 16.2 (Line Integrals): #3, 5, 8, 11, 13, 19, 21, 33, 39, 40, 41.
• Section 16.3 (The Fundamental Theorem for Line Integrals): #3, 4, 6, 7, 9, 12, 14, 15, 18, 19, 20, 21, 22.
• Section 16.4 (Green's Theorem): #3, 4, 5, 7, 8, 11, 13, 17, 18, 19.
• Section 16.5 (Curl and Divergence): #2, 4, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 21, 22, 24.
• Section 16.6 (Parametric Surfaces and Their Areas): #3, 5, 6, 20, 23, 24, 34, 35, 38, 42, 43.
• Section 16.7 (Surface Integrals): #6, 11, 12, 14, 16, 19, 21, 24, 25, 29, 37, 38.

My office hours this week are Tuesday/Thursday 4-5:30pm. The tutoring room will be definitely be open Monday, Tuesday and Wednesday. If there are additional hours I will post them here.

The cumulative final exam for section DL1 will be held Wednesday, May 9, 8:00 AM – 11:00 AM in 314 Altgeld Hall. Here are the sections covered.

• Test 1 material (12.1, 12.2, 12.3, 12.4, 12.5, 12.6, 13.1, 13.2, 13.3, 13.4, 14.1, 14.2, 14.3, 14.4, lines/planes worksheet)
• Test 2 material (14.5, 14.6, 14.7, 14.8, 15.1, 15.2, 15.3, 15.4, 15.5, 15.6)
• Test 3 material (15.7, 15.8, 15.9, 16.1, 16.2, 16.3, 16.4, 16.5, 16.6, 16.7)
• Additional material (16.8, 16.9)

See the list of remaining tutoring/office hours for the semester.