## CALCULUS III ONLINE

This is Razvan Gelca's webpage dedicated to teaching Calculus III during the Coronavirus pandemic. I am Razvan Gelca, I am a Full Professor of Mathematics at Texas Tech University. I was born in Romania and I have moved to Texas right when the third millenium started. I love teaching differential and integral calculus, within my best-seller problem book Putnam and Beyond more than 100 pages are devoted to calculus problems. For a very long time I have been one of the coaches of the US International Mathematical Olympiad Team, which is currently the best IMO team in the world. I have worked with students between sixth grade and doctoral level, I have thirty years of teaching experience, and I will do my best to guide your learning of calculus in these times of crisis. I really miss being in the classroom, and I will use this web page to communicate with you.

Before you proceed, please read the syllabus, which can be found here.

I have written detailed instructions about how to study, do the homework, and take quizzes, and exams. These instructions can be read here.

### WHAT TO LEARN

As the class progresses, I will give you a week-by-week outline, with details of what to study. Each week is indexed by its dates. Please make sure you follow the weekly schedule as described.

• 08.24-08.28.2020. You should review 9.1, 9.2, 9.3, and 9.4. From those sections you should know the definition of vectors, the notation (including the use of i,j,k), and the definition of the dot product and cross product.
Read the sections 9.5, 9.6, 9.7. Focus on
• Section 9.5: the parametric equation of a line, Example 1, parametric equations, Example 5, Example 6, Example 7, parametrizing a curve, Example 8,
• Section 9.6: equations of a plane, Example 1, Example 2,
• Section 9.7: catalog of quadric surfaces, know how to identify a quadric (Example 4), and how to solve problems 3-14 at page 740 (this is not a homework, just a mental exercise for you to prepare for later material).
There is one homework associated to this material (HWK1).

• 08.31-09.04. We finally start differential and integral calculus, with Chapter 10. This chapter models trajectories in the 2-dimensional and 3-dimensional space, as functions from the real numbers to the 2- or 3-dimensional space. You should think that $${\bf F}(t)$$ is the location (in the plane or the space) at time $$t$$ of some particle that travels by the specified formula. In studying this chapter you should focus on the following things: There are also video lectures produced by the chair of the Department of Mathematics and Statistics, which can be found here: Section 10.1, Section 10.2-10.4, I, Section 10.2-10.4, II, Section 10.2-10.4, III. There is one homework and one quiz associated with this material.

• 09.08-09.11. You can also watch these videos: video 1 and video 2.

• 09.14-09.18. This week we learn some concepts related to partial derivatives.
• In Section 11.4 focus on the equation of the tangent plane at a point, Example 1, incremental approximations, Example 2, total differential, Example 4. Here is one video.
• In Section 11.5 focus on the chain rule, Examples 2,3, with this video and implicit differentiation, Examples 4 and 5, and this video.
• In Section 11.6 focus on directional derivatives, Example 1, gradient, Example 2, Example 3, basic properties of the gradient, Theorem 11.10, Example 4, Example 5, Theorem 11.11, Example 6, Example 7, tangent planes and normal lines, Example 9, Example 10. There are videos here, here, here, here, here, and here.
For all these sections there are also these videos:video 1, video 2, video 3, video 4, created by our chair.

• 09.21-09.25. This week you learn how to apply differential calculus to find extrema. This is one of the great successes of calculus. For another discussion of these click here where the discussion is towards the end, and here.

• 09.28-10.02. Because this week there is an exam, we will cover less material, so your homework will be shorter. We start multiple integration.
• In Section 12.1 read the introduction and focus on Fubini's theorem (Theorem 12.2) and its applications in Examples 2,3,4. You can look at the following video and I recommend more this one. An application is here. All these are pretty straightforward.
• The new stuff, where you have to think more, is in Section 12.2. Read everything in that section, the look at the following videos video 1, video 2, video 3, video 4, and video 5. We will see more of this later.
There is also this video you can watch.
There is one homework but no quiz associated with this material. There is one midterm exam this week.

• 10.05-10.09. We continue with the computation of integrals.
• I will give you two more exercises for Section 12.2. Here are a few more videos: video 1, video 2, video 3, video 4. The last two are about the change of order of integration.
• In Section 12.3 you learn how to change variables. This section is about switching to polar coordinates. You need to learnd the statement of Theorem 12.4 and read its applications from Examples 1,2,3,4,5. Here are some videos: introduction, example, example, computation of a volume.
• Section 12.4 shows you how to compute the area of a surface. The formula for area if the surface is the graph of a function is on page 952 in the box, with Example 1 and Example 2 as applications. There is also a formula for the area in the box on page 956, when the surface is in parametric form. Example 4 is an application. There are videos here, here, here, and here.
• For this week there are also videos here and here. This week there is one homework and one quiz.

• 10.12-10.16 This week we learn triple integrals. The main idea is the same as with double integrals, except that we integrate with respect two three variables and not just two. Computations can be quite unpleasant. There is also these video: here. This week there is one homework and one quiz.

• 10.19-10.23. We start Chapter 13. This is the most difficult, so we will take it slowly. This week we cover two sections.
• Please read the entire Section 13.1. It contains important explanations and definitions. There are some videos you can watch: video 1, video 2, video 3.
• In Section 13.2 read Example 1, Example 2, understand what is in the yellow box on page 1031, then read Examples 4, Example 5, Example 6, the brown box at page 1036, adn Example 8. There are videos too: video 1, video 1, video 3 video 4 video 5.
There is a video also here. This week there is one homework and one quiz.

• 10.26-10.30. We finish the story about line integrals and start talking about integrals on surfaces.
• In Section 13.3 learn Theorem 13.2 with Example 1, then the definition of conservative vector fields and Theorem 13.3, with Examples 3 and 4, and Theorem 13.4 with Example 5. Finally, Theorem 13.5 and Example 7. There are some videos here, here, here, and here.
• In Section 13.4 we learn about integrals on the simplest surfaces: domains in the plane. Read the explanation before Theorem 13.6 and then learn the theorem itself. Read Example 1, 2, Theorem 13.7 and Example 3. Here are some videos: video 1, video 2, and video 3.
These sections are also covered here and here (the second video also covers 13.5 which we only need next week). This week there is one homework and one quiz.

• 11.02-11.06. This is the last full week of new material, then we have only half a week left of new stuff. We are generalizing Green's formula, and for that we need surface integrals.
• In Section 13.5 you should read the formula in the yellow box at page 1064 and Example 1, then the formula in the box at page 1065, the formula in the box at page 1067, and Example 3, Example 5 and finally Example 6. Videos for some of this stuff are here, here, here, and here.
• In Section 13.6 you should know Theorem 13.8, then read Example 1, Example 3. Watch this and this.
• You can also watch this video. This week there is one homework and one quiz.

• 11.09-11.13. Good news! We are done with learning new material. One more section. This is Section 13.7. Learn Theorem 13.9 and Examples 1, 2, 3. Watch the videos: first, second. This week there is one short homework and one quiz.

### HOMEWORKS

You can find information about WebWork here and about how to enter homework here.

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### QUIZZES AND EXAMS

Quiz 1 (due Friday, 09.04.2020 at noon): Find the unit tangent vector $${\bf T}(t)$$ and the principal unit normal vector $${\bf N}(t)$$ for the curve given by \begin{eqnarray*} {\bf R}(t)=3t^2{\bf i}+2t^3{\bf j},\quad t\neq 0. \end{eqnarray*} Click here for the answer.

Quiz 2 (due Friday, 09.11.2020 at noon): For the curve given by \begin{eqnarray*} {\bf R}(t)=(\sin t){\bf i}+(\cos t){\bf j}+t{\bf k} \end{eqnarray*} find the curvature when $$t=\pi$$ and the length of the curve from $$t=0$$ to $$t=2\pi$$.
Quiz 3 (due Friday, 9.18.2020 at noon): Using implicit differentiation, find $$dy/dx$$ for the function $$y$$ defined implicitly by the equation \begin{eqnarray*} e^{xy}+x\sin y=3. \end{eqnarray*}
Quiz 4 (due Friday 9.25.2020 at noon): Find the absolute extrema of the function $$f(x,y)=3x^2+2y^2$$ in the region $$x^2+y^2\leq 4$$.
Exam 1 has been posted here. The solutions are now posted here.
Quiz 5 (due Sunday 10.11.2020 at 3pm) Find the area of the portion of the paraboloid $$z=x^2+y^2$$ that lies inside the cylinder $$x^2+y^2=4$$.
Quiz 6 (due Friday 10.16.2020 at 3pm) Evaluate the integral \begin{eqnarray*} \iiint_D (x^2+y^2+z^2)dxdydz, \end{eqnarray*} where $$D$$ is defined by $$x^2+y^2+z^2\leq 2$$.
The solution is here.
Quiz 7 (due Friday 10.23.2020 at 3pm) Find the work done by the force field $${\bf F}=(2x^2+2y^2){\bf i}+(3x+3y){\bf j}$$ as an object moves counterclockwise along the circle $$x^2+y^2=4$$ from $$(2,0)$$ to $$(-2,0)$$ and then back to $$(2,0)$$ along the $$x$$-axis.
The solution is here.
Quiz 8 (due Friday 10.30.2020 at 3pm) Show that the vector field $${\bf F}(x,y,z)=yz^2{\bf i}+xz^2{\bf j}+2xyz{\bf k}$$ is conservative, find a potential function $$f$$ and evaluate $$\int_C{\bf F}\cdot d{\bf R}$$ where $$C$$ is a smooth path from $$(1,0,1)$$ to $$(2,2,3)$$.
The asnwer is here.
Quiz 9 (due Friday 11.06.2020 at 3pm) Compute \begin{eqnarray*} \oint_C (ydx +zdy+xdz) \end{eqnarray*} where $$C$$ is the triangle with vertices $$(3,0,0)$$, $$(0,0,2)$$, and $$(0,6,0)$$ traversed in this order. The solution is here.
Quiz 10 (due Saturday 11.14.2020 at 12.01pm, so that we avoid the Friday 13 due date) Use the divergence theorem to evaluate $$\iint_S {\bf F}\cdot {\bf N}dS$$ for $${\bf F}=xyz{\bf j}$$, where $$S$$ is the cylinder $$x^2+y^2=9$$, $$0\leq z\leq 5$$ with the two disk ends included. The solution is here.
Exam 2 has been posted here. The solutions are now posted here.

The Final Exam has been posted here. You have until next Tuesday at 1 pm to solve it. Exactly 7 days!
End of page.