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 bestseller 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 weekbyweek outline, with details
of what to study. Each week is indexed by its dates. Please make sure
you follow the weekly schedule as described.
 08.2408.28.2020. 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 314 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).
 08.3109.04. We finally start differential and integral calculus, with
Chapter 10.
This chapter models trajectories in the 2dimensional and 3dimensional
space, as functions from the real numbers to the 2 or 3dimensional 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:
 Section 10.1: The definition of vector valued functions, Examples
1 and 2, vector function operations, Example 4. It might be worth
reading the explanation of limits and continuity.
A good explanation video is here.
 Section 10.2: The derivative of a vector valued function, Example 1,
Example 2, the concept of a tangent vector, Example 3, deciding
when a curve is smooth, Example 4, rules for differentiation,
Example 6, Example 7, Theorem 10.4, modeling motion, Example 8,
vector integrals, Example 9, Example 10. I recommend several videos:
the derivative,
properties of the
derivative, derivative of the cross product, smooth curves, modeling velocity and acceleration, finding the indefinite integral, integration with initial condition (velocity and acceleration).
 Section 10.4: Unit tangent and principal unit normal vector, Example 1, (the rest of the section will be studied next week).
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 10.210.4, I,
Section 10.210.4, II, Section 10.210.4, III.
There is one homework and one quiz associated
with this material.

09.0809.11.
 Let us finish 10.4 with arc length and curvature (watch
this videos: video 1,
video 2,
video 3,
video 4 . Then
proceed with Sections 11.1 and 11.2 (just read them, you will understand thing
better). Then we continue with

Section 11.3. Focus on partial differentiation, Example 1, Example 2, Example 3,
Example 4, Partial derivative as slope/rate of change, Example 5, Example 6,
higherorder partial derivatives, Example 7, Theorem 11.1, Example 8, Example 10. I recommend the following videos: First order partial derivatives, Implicit differentiation, Second order partial differentiation.
You can also watch these videos: video 1 and
video 2.
 09.1409.18. 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.
 09.2109.25. This week you learn how to apply differential calculus
to find extrema. This is one of the great successes of calculus.
 In section 11.7 read the theory and all examples. All is
very important. It might be that the least squares method
looks a bit hard, I will not test it, but you might see it again
in your future, just read it even if you do not understand it.
Watch the following videos: How to find extrema, applications 1, applications 2,
applications 3, finding extrema on domains that have boundary.
 From 11.8 the method of Lagrange multipliers, look at Theorem 11.15
and the method described immediately after, then at Examples 1 and 2.
There are three videos: two variables, three variables, and one example
For another discussion of these click here where the
discussion is towards the end, and
here.
 09.2810.02. 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.
 10.0510.09. 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.
 10.1210.16 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.
 In Section 12.5 read Theorems 12.5 and 12.6 and then Example 1, 2, 3, 4, 6. Here are some videos: volume 1, volume 2, volume 3,integral
 In Section 12.7 you learn cylindrical and spherical coordinates. Learn
the four "yellow boxes" from the theory then Example 2, Example 5, and
Example 7. Here are some videos cylindrical coordinates, cylindrical coordinates, cylindrical coordinates, spherical coordinates, spherical coordinates.
There is also these video: here. This week there is one homework
and one quiz.
 10.1910.23. 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.
 10.2610.30. 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.
 11.0211.06. 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.
 11.0911.13. 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
Click here to start doing homework.
You can find information about WebWork here and about how to enter homework here.
Messages: The first homework is now closed.
The second homework is now closed.
The third homework is now closed.
The fourth homework is now closed.
The fifth homework is now closed.
The sixth homework is now closed
The seventh homework is now closed.
The eighth homework is now closed.
The ninth homework is now closed.
The tenth homework is now closed.
The eleventh homework is now closed.
The twelth homework is now open.
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 \(t=2\pi\).
Click here for the answer.
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 \(x^2+y^2\leq 4\).
Click here for the answer.
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.