Here are a selection of Students’ Study Tips. Feel free to post comments and more tips in the comments section.
On saving time integrating
“When taking a double integral of a function f(x,y) dx dy. If you can write f(x,y) as g(x)h(y), then you can take the integrals separately. That is, f(x,y) dx dy = g(x) dx h(y) dy. It can save a lot of time!” – Huge
[EDIT: This is a great suggestion, but you have to be careful about the limits. For example, consider 
. According to this suggestion, you can turn it into 
. But you can’t change it into 
because the latter won’t simplify to a number! ]
On surface integral formulas
Anonymous was pleased to clear up the following confusion about surface integrals. The surface integral formula
becomes
when you choose your parametrisation. At first, this seems wrong, because the first formula appears to have a unit normal, while the second does not. But in fact, this is because 
becomes 
, so we observe some cancellation:
On the meaning of “closed”
When I say “closed” in this class, I mean “has no boundary,” and this occurs in two situations:
A “closed loop” is a curve with the same endpoint as starting point.
A “closed surface” is a surface which cuts out a finite piece of 3D space, i.e. has a finite inside distinct from its outside. This also has no boundary or “edge”.
You may also come across the term “closed” in mathematics to mean “complement of an open set.” (You have probably heard of a “closed interval” in this sense.) In this course we don’t talk about open sets much, and I don’t think we’ve talked about closed sets at all. So when I say closed, I am speaking of a curve or a surface, and I mean that it has no boundary.
— Kate (channelling Pippi Longstocking)
On visualising graphs
“Coming from a Math background and not a physics background, I always had trouble visualizing graphs in 3-D. I usually just refused, put it off, or tried to graph it online. Sometimes I would even plug in number after number until I somewhat could visualize what it looked like. It seems extremely trivial now, but I learned to cover up one of the variables and graph only the other 2, and then repeat this process for the other two combinations. So, take F=<P,Q,R>, I would cover up the P value and know that I am looking down the x-axis onto the yz plane (this is what I never understood before; the fact that I was looking down the axis that I was covering up). I would graph this and then repeat by covering up the Q value, which corresponds to looking down the y-axis onto the xz plane, and the R value which corresponds to looking down the z-axis which looks onto the xy plane. Once I had a graph for each plane (xy, xz, and yz), it became much easier to put it all together and see the graph in 3-D. It’s so easy now but has honestly saved me so much time and frustration! It has also helped me get a better understanding of calculus in general as I am more capable of visualizing what happens to surfaces. ” – Anonymous
On remembering the curl formula
“I always had trouble with how to remember which P-partial went with which R-partial, etc., when we’re trying to find 3-D vector field is conservative by equating certain partial derivatives. So, I finally realized there was a trick to remember this:
I first start out by writing, PQR in a column on the top of my exam, since they’re in alphabetical order. In the next column, shift PQR down by one spot, to obtain QRP
P = Q
Q = R
R = P
Then, the variable you’re taking the partial with respect to corresponds with where the corresponding variable in the column to the right is, looking at the vector equation.
For example, thinking of <P,Q,R> as <x,y,z> :
Py = Qx (since Q is in the y-spot and P is in the x-spot)
Qz = Ry (since R is in the z-spot and Q is in the y-spot)
Rx = Pz (since P is in the x-spot and R is in the z-spot)
Then, to check, you should have the capital letters (field components) in alphabetical order on the left side, and the ‘with respect to’ variables in alphabetical order on the right side. :)” — Anonymous
A useful website (from Anonymous)
http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx
Curl and divergence: which is which?
“At first, I keep messing up Curl and Divergence. I couldn’t remember which uses cross product and which uses dot product. After a while, I remembered it by thinking that Curl starts with a ‘C’ like ‘C’ross product and Divergence starts with a ‘D’ like ‘D’ot product.It’s that simple. =)” — Anonymous
Orientation of surfaces with several parts
“If you are integrating on a surface that consist of a least two different pieces, then the orientation of each integration has to agree with the others. Pay special attention if the question is asking for outward orientation instead of upward. For example, a hemisphere with its bottom disc will have normal vectors that are pointing in different directions for the two pieces. For the part of the hemisphere, its normal is pointing more upwards, but for the bottom disc, it’s pointing downward. ” — Jimmy Wales’s Personal Appeal
On prerequisites and review
“To do well in this course, I would HIGHLY recommend doing a thorough review of 3D surface/volume parameterizations. While it has been a year since I took MATH 253, I felt as though I remembered enough of it to be able to get by. I soon found out, however, that much of the material that is covered in MATH 317 draws heavily on what was taught in Multivariable Calculus, and it would have been wise for me to ensure that my understanding of the material from that class was solid. Much of my trouble with the material in this course isn’t a result of the new concepts being difficult to grasp, but rather because I am not as familiar with material that was learned in previous courses.” — Anonymous
On choosing the orientation
“When computing for the normal sometimes it is confusing to set the order of the vector multiplication. Just cross product the two vectors in whatever order and then later on visualize the actual situation or graph and set the sign accordingly without thinking about it too much during the calculation.” — Anonymous
or 
), but don’t memorise the constants!
is NOT conservative, because if it were, then all line integrals on closed loops would be zero, and we found a counterexample in part (c). The fact that 
would usually tell us that 
and 
, which is a piece of the surface lying over the 1st quadrant of the xy-plane. This surface is a parabolic bowl, facing downward, and shifted up, and we are asked to use the part of the bowl above the xy plane. This piece looks like an upside-down bowl. However, the solutions give a domain that would give just the noon-to-3-o’clock portion of the bowl. If you have any doubts, please graph the two parametrised surfaces. (In particular, if we dipped the bowl down into a sink of water, it would hold air. But the piece of surface over the domain the solutions suggest wouldn’t hold any air!) This error is an excellent learning opportunity — please do investigate what I’ve described in this paragraph.
is, it is parallel to itself, and parallel vectors have zero cross-product.
. 
.
.
for any field with a hole in its domain. That’s not true: the integral must depend on the vector field. Not all vector fields with a hole at the origin have integrals 
is the upper hemisphere (
) of a unit sphere oriented upward, and 
is the unit disk in the 
-plane oriented upward, is 
?
. Would you think I was lying? If not, what could you conclude about the divergence of 
. In other words, if 
, it must be that 
. That is, on simply connected domains, all fields with zero curl come from the gradient of a function (i.e. are conservative).
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