Multivariable Calculus Lecture Ideas I am teaching a course in multivariable calculus this semester.  We are covering the basics about $\mathbb{R}^n$, including dot products and cross products, curves, and quadric surfaces.  After that we learn differentiation for functions $f:\mathbb{R}\rightarrow \mathbb{R}^n$ and partial differentiation for functions $f:\mathbb{R}^n\rightarrow\mathbb{R}$.  We finish off with Lagrange multipliers.  
My program so far has been to lecture on a chalkboard while they take notes.  I am not really keeping their attention.  I was wondering if anyone had any ideas to involve them more in class.  We are in a computer lab, and have access to Maple.  Your answer need not be computer based, but I have computers available.
 A: You could try to implement Eric Mazur's Peer Instruction. There is a great video about it here, and here are two articles discussing how it works. The latter has links to statistical evidence that students learn better with this method (for me, the jury is still out). I guess the basic idea is that classes start by asking students a multiple choice question, and they vote on it using clickers. It should be something controversial, about a conceptual point rather than a numerical computation. After the first round of voting, students talk to each other and try to convince one another of which is the correct answer. The class then votes again. The idea is to spend class time on absorbing the material rather than hearing it for the first time. It's expected that students will have done all the reading on their own before class. There are links on Mazur's website which are designed to help professors implement this. A computer lab should have everything you need.
A: Here is another answer, based on the way your question was phrased. It seems you're looking for ways to effectively use the computers in the room. David Perkinson from Reed College is great at this, and generally lectures from slides with images created in mathematica or maple. His slides can be found here and I imagine he would send you tex code or image files if you wanted. One very clever idea he uses is to leave blanks in the versions of the slides he gives to the students so they have to write on them. That way they have to listen actively, but also get to see things in a way which a textbook or blackboard might not be able to show.
A: When introducing the cross product, one thing that always catches my students' attention is a demonstration of precession. We have a bike wheel with a heavy solid rubber tire. (You may be able to borrow one from your physics department.) I get it spinning with its axis in the horizontal plane, and then suspend it by a rope from one side. They are gratifyingly surprised when they see it precess in the horizontal plane rather than flopping. This connects to torque expressed as a vector cross product.
Another example I like to do is to have them imagine a spinning cylindrical space station and lead them through a series of thought experiments leading them to figure out the rules by which the centrifugal and Coriolis forces operate. E.g., a person standing on the deck releases a ball. In the frame of the stars, the ball travels straight. What appears to happen in the rotating frame? Once they see that the centrifugal force operates like a radially directed gravitational field, I do examples that show the existence of a velocity-dependent force. E.g., what happens if the person standing on the deck throws the ball opposite to the direction of rotation, such that he exactly cancels its motion in the frame of the stars? What if a ball is released on the axis? What if the ball is released on the axis with some small radial velocity? Based on this sort of thing, it's fairly easy to get them to conclude that the Coriolis force is an odd function of the ball's velocity, and also an odd function of the angular velocity of the space station. There is only one good way to get a vector-valued function with these properties, so it must be $\mathbf{F}=c\mathbf{\omega}\times\mathbf{v}$. They enjoy imagining the fun and strange things that happen on the space station, and it's a cool example where the uniqueness of certain mathematical operations makes it possible to determine the answers to problems without grotty calculations.
A: I used this more for integration and change of variables, but still you might be able to use the idea.
I took my TA section outside to a (ground level!) skylight which was a mostly transparent spherical
cap with some gridlines on it.  It was sunny, and you could peer down to see the shadow of the lines
on the floor below.  I used this as an example of how projection might change the area, and how
acting on each area element needed to be considered in doing the integration.  You might be able
to come up with a similar visual aid for the chain rule.
Gerhard "Ask Me About System Design" Paseman, 2012.09.19
