Well ... since we elected to re-boot today instead of going over new material, we're not quite as far as I wanted to be. That is OK, though, since I explicitly budgeted extra days just in case we wanted to spend more time on math background.
What this means is that on Wednesday, we'll start out by covering vector derivatives (div, grad, curl) and partial derivatives, and probably get to line integrals and the remainder of the vector calculus we'll need on Friday. By the end of Wednesday's homework, you should know how to do all but the very last of Friday's homework problems.
Homework 1 has 9 problems: 1-4 are due Wednesday, and 5-9 are due Friday (in both cases, by the end of the day). Here are some stray hints on the first 4 problems:
(1) The 'separation vector' is just the result of subtracting the two vectors. A unit vector can be constructed by dividing any vector by its magnitude.
(2) Body diagonal = opposite corners of a cube; there are four possibilities. Pick any two, the angles are the same. You could define two of these as (1,1,1) and (1,-1,1) for instance. If it helps, draw two cubes stacked on top of one another ...
(3) Just grind through it ... easy but tedious. A good chance to practice the cross-product-as-determinant rule! You can double-check some of your answers by verifying that axb is perpendicular to both a and b (meaning a-dot-b is zero).
(4) The first and last are not so bad, just remember how to take the derivative of a vector function, and that a and b are constant vectors. For the second, keep in mind that a and b are not necessarily perpendicular, so your normal rule for taking cross products does not work. Either you need to write the a and b vectors in a cartesian basis (in terms x & y unit vectors, say a=a_x xhat + a_y yhat), or you need to define a unit vector perpendicular to both a and b as well as the angle between a and b.
Probably the former option, writing a and b in terms of cartesian components, is easiest. This is still general, since you have the freedom to pick the x-y plane as the same plane formed by a and b for any two vectors a and b. You should not need to do this for the first and last parts, however.
Wednesday, we'll go over problems 5-9 a bit so you know how to get started.
No comments:
Post a Comment