The first group activity, discussed in more detail in Part II, asks the students to draw a particular vector, expressed with respect to 2 different orthonormal bases. We call two students to the board to present their answers, one for each basis, but on the same diagram. Frequently, the second student uses a different scaling than the first; the resulting vectors differ. On several occasions, it has taken several attempts to get this student to realize that this is wrong! Such students are clearly viewing the results of the two parts of the problem as completely different problems, and not immediately seeing any reason that the answers should have anything to do with each other!

A good problem involving the cross product is to show that the sum of the “area vectors” of a tetrahedron is the zero vector [ 3 ] Each area vector is (one-half) the magnitude of the cross product of two sides; the problem requires the student to realize that the three vectors in a triangle add to zero (if appropriately oriented), and to use the algebraic properties of the cross product.

An Honors College student did this problem correctly for homework, then gave a more “detailed” proof using components — correctly, using arbitrary components for all vectors. This student clearly felt that an argument without components was somehow suspect!

First and foremost, physics is about physical quantities, such as temperature. What matters is the temperature here, not the functional representation of the temperature in terms of some coordinate system. This corresponds more closely to the differential geometer's notion of a scalar field than to the notion of a function as traditionally taught in a calculus course.

It is essential to know what sort of object one is dealing with. Temperature is a scalar field, but force is a vector field. Lengths have different units than areas — causing many physicists to cringe when confronted with an equation such as $y=x^2$! And time is special; using $t$ as a parameter is only a good idea if a problem is dynamic, not static. (See Chapter 8.)

Vectors are arrows in space, not pairs or triples of numbers. Components may — or may not — be useful when computing things, but the use of a particular basis or coordinate system is a matter of convenience. The geometry of the dot and cross products, as projections and areas, respectively, is at least as important as the ability to do the algebra necessary to compute them. (See Chapter 9.)

Elementary physics takes place in three dimensions. But it is not possible to graph the temperature in a room; that would require four dimensions. So emphasizing graphs as the fundamental way to visualize functions is not helpful in physics. In particular, overemphasizing 3-dimensional graphs of functions of 2 variables is not appropriate, as this skill does not carry over to functions of 3 variables. It is much more important to convey that functions associate values with points, which can be done very nicely using contour diagrams or color. (See Chapter 10.)

A related issue is that the range and domain of a physical quantity such as temperature usually have different dimensions. Using the same units along both axes is very misleading, as is the exclusive use of $x$ and $y$ as labels. For this reason, hills are not typical examples of functions of 2 variables.

Undergraduate physics and engineering involves straight lines, circles, planes, cylinders, and spheres. And that's about it. No parabolas; no paraboloids. Yes, it's nice to have techniques to handle the general case, but not if they obscure the highly symmetric examples students will see in their subsequent coursework. Interesting physics problems can involve trivial mathematics!

When dealing with a round problem, physicists will work in round coordinates. Be careful! When a physicist switches to, say, polar coordinates, a simultaneous switch to the corresponding basis vectors is implicitly understood. Yet many mathematicians have never even heard of, say, $\rhat$, the unit vector in the radial direction! (See §3.2 of Chapter 3.)

There's an old joke about what physicists and mathematicians do when confronted with a problem they can't solve. The physicist makes some approximations to get a reasonable answer. The mathematician changes the problem to one which is solvable.

First of all, physics problems don't come with preferred coordinates, or a list of independent variables. Curves and surfaces don't come with parameterizations. Unknowns don't have names. Getting to a well-defined math problem is part of the problem — often the hardest part!

Going the other way, if you can't take a math problem and add units to it, it's a poor physics problem. (See §8.3 of Chapter 8.)

Yes, students must have some basic skills. But “template problem-solving” is not one of them. Rather, they need a few key ideas, which they can remember.

Solving problems is not straightforward. Physics often involves the interplay of multiple representations, such as being able to compute dot products both algebraically and geometrically in the same problem. (See the example at the end of §9.1 of Chapter 9.)