 Fall 2002 |
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Fall 2002 |
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| Send questions of potential general interest to math21afaq@math.harvard.edu. |
| Question: I just had a quick question regarding the challenge problems posted on the Math 21a web site: Are these problems extra credit or are they simply intended to supplement and reinforce the chapter questions? Thanks. |
Answer:
Doing challenge problems is one of several options for a section project:
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| Question: Why can the z-coordinate be anything when it is not specified? Example: in the equation x2+y2=1, the variable z does not appear. | Answer: You are only given constraints to the possibilities of x and y. The z value can be arbitrary. In the example above, the points satisfying the equation are a circle in R2, but since any z value will still allow x and y to satisfy the conditions, in R3 you actually get a cylinder. Just like in single variable calculus, you are only given constraints and must graph any points/lines/planes/surfaces that are able to satisfy those constraints. |
| Question: What really is the difference between <1,1> and (1,1)? I know one is a vector and one a point, but don't they signify the same thing? | Answer: A vector and an origin define a point. So the vector <1,1,1> extends from the origin (0,0,0) to the point (1,1,1). What is key here is that a well defined (and consistent) origin has been selected. Comparing vectors does not require an origin, but comparing points or measuring distances does. In general, a well defined origin is assumed for this course, so vectors can often be used to signify points. |
| Question: I can't keep all the vectors and scalars straight--if you multiply two together, what do you get? |
Answer:
There are a number of things that can occur when multiplying vectors and scalars:
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| Question: In number 34, section 9.2, does the |r-r1| denote absolute value or magnitude of the vector? | Answer: They represent the magnitude (length) of the vector going from r to r1. That is, the question is asking you to describe a constant sum of distances. |
| Question: Why is the gradient of f(x,y) not perpendicular to the surface defined by f(x,y)? | Answer: The gradient of f is perpendicular to the level sets of f, not the surface. An easy way to see that is the following: the gradient of g(x,y)=x+y is the vector (1,1) in the x-y plane. This is not always perpendicular to the plane 0=x+y-z but it is perpendicular to the level set z=0 of that plane (which is a line in the x-y plane, namely the intersection of the given plane with the plane z=0). |
| Question: When do I use LaGrange? When do I plug in the boundary? When do I just take the gradient and set equal to zero? | Answer: When you are finding extrema of a function, there are two cases, one called bounded and the other unbounded. If there are no restrictions on a function (no other conditions it must satisfy) then simply take the gradient and set equal to <0,0,0> to find the critical points and then check them using the test (if D<0 and fxx>0 etc). If you are given a simple boundary, like a square, you do the above, check to make sure that your answers are within the boundary and then plug in the boundary equation(s) to check points there (this process is similar to solving a system of equations, but don't forget the corners!). Finally, if you are maximizing a function subject to another equation or boundary surface that is somewhat complicated (like an ellipsoid), you can use LaGrange multipliers to find the max value on the surface (but you might have to check the inside of the surface, so be careful--a way to notice this is if the constraint equation has something other than an equal sign). |
| Question: What does stretching factor mean and when do I need them? | Answer: Th "stretching factor" is a name used to denote something you add when switching variables in integration. It is related to the fact that you are no longer integrating over a rectangle (like you would in 'rectangular' coordinates). All you need to know for 21a is that if you switch to polar you need to change dxdy to rdrd(theta); to cylindrical dxdydz becomes rdzdrd(theta); to spherical dxdydz becomes (rho)sin(phi)d(rho)d(phi)d(theta). When you parameterize for surface integrals and the like, you do not need a stretching factor. The reason for this is the fact that in the parameter space, you are still integrating over rectangles so there is no change. (this is not necessary to understand for any test.) |
| Question: What is a vector field? | Answer: It is a relationship that gives a vector for each point in the vector field. That is, for each point (a,b,c), there is a vector [p(a,b,c),q(a,b,c),r(a,b,c)]. This has many physical interpretations but that easiest is thinking of it as a description of the wind. If you start at (a,b,c), you are "pushed" by the vector defined by those points to another point, and then pushed again, etc. This is why it is often refered to as a force field (that "moves" objects). |
| Question: When do I switch from rectangular to polar, cylindrical, or spherical? | Answer: When either the function that you are integrating or the area over which you are integrating is easier in different coordinates. For instance, if you are integrating over a sphere, or if you are integrating the function x2+y2-z it would probably be a good idea to switch. Sometimes they are more complicated, and it takes practice to figure out which is easiest and then what the easiest order of integration is. |
| Question: Is there an easy order of integration for non-rectangular coordinates? And, when can you simply switch the order? | Answer: No. But, the best choices are drd(theta), dzdrd(theta), and d(rho)d(phi)d(theta), usually. There are many exceptions, however, but a good rule is to keep the theta last, as you are usually going around in a whole circle or a fractional part thereof. If all of the limits are numbers, than you can simply switch the order of integration. If some limits depend on variables, though, you have to be careful because once a letter has been integrated, it can't appear in the limits. See the section in the book or review homework/tests to get some practice switching between limits when different variables depend on each other. |
| Please send comments to math21a@fas.harvard.edu |
