Country Boy

Applying the differential operator [math]\frac{d}{dx}- k[/math] to the function [math]e^{kx}[/itex] gives you 0: Applying the differential operator [math](\frac{d}{dx}-k)^2[/itex] either of [math]e^{kx}[/math] or [math]xe^{kx}[/math] gives 0. In general, applying the differential operator [math](\frac{d}{dx}-k)^n[/math] to the n functions, [math]e^{kx}[/math], [math]xe^{kx}[/math], [math]x^2e^{kx}[/math], ..., [math]x^{n-1}e^{kx}[/math] gives 0. You should be able to prove that by induction.

Yeah, sorry, I just skipped over this: Well, if you include the direction, i.e. distinguish between "parallel" (same direction, angle 0) or "anti-parallel" (opposite directions, angle pi) you can't. Of course that is true of the cosine as well. If two vectors have cosine of the angle between them, AND YOU INCLUDE "DIRECTION" WITH "ANGLE" we could not tell whether the anglee between them is pi/2 or 3pi/2. You need both sine and cosine to decide exactly what the angle and directions are.

The two "x"s are different. In writing g(f(x))= x, yes, the assumption is that x is in the domain of f, the set A. In writing f(g(x))= x, the assumption is that x is in the domain of g, the set B. It would have been more precise to say "For all x in A, g(f(x))= x and for all x in B, f(g(x))= x". I, personally, would probably have said "For all x in A, g(f(x))= x and for all y in B, f(g(y))= y" to make the distinction even clearer.

I said: "Strictly speaking the cross product is only defined for 3 dimensional vectors, but the cross product of the two vectors <ax, ay, 0> and <bx, by, 0> is the vector with length axby- aybx pointing along the z-axis." In other words, that is routinely used as the LENGTH of the cross product of two vectors in the xy-plane. As I said, the true cross product would point along the z-axis, in the positive or negative direction depending on the order of the terms- the cross product is ANTI-commutative. Atheist wasn't talking about the angle between the two vectors, he was talking about the direction of their cross product. If the angle between two vectors is 0, since sin(0)= 0, their cross product is the 0 vector. It has NO direction.

The CROSS PRODUCT of two vectors u x v is defined as the vector with length |u||v|sin(angle) pointing in the direction perpendicular to both u and v, in the "right hand rule" direction (Curl the fingers of your right hand from u to v. The direction your thumb points is the direction of the product vector.) Strictly speaking the cross product is only defined for 3 dimensional vectors, but the cross product of the two vectors <ax, ay, 0> and <bx, by, 0> is the vector with length axby- aybx pointing along the z-axis. The length of the cross product of vectors u and v is, also, the area of a parallelogram having vectors u and v as adjacent sides (even in 3 dimensions). The "triple product" u.(v x w) is the area of a parallelopiped having vector u, v, and w meeting in one vertex.

A sailboat is a wind powered vehicle!

I responded telling how you can calculate 10^1.5 (it's about 31.6). If your question was how to convert the "exponential" equation, 10^1.5= 31.6, to a "logarithmic" equation, the answer is exactly what you did with 2^3= 8. The base of the exponent, here 10, becomes the base of the logarithm, the "result", here 31.6, becomes the argument of the logarithm, and the argument of the exponential becomes the "result" of the logarithmic equation: 10^1.5= 31.6... becomes log_10(31.6...)= 1.5. As dave said, the exponential equation a^x= y becomes log_a(y)= x. Since "exponential" and "logarithm" (to the same base) are INVERSE functions, changing from one to the other just swaps "argument" and "result" (x and y).

There was no "objection". What I said was that the FUNCTION f(x)= (cos(x), sin(x)) is transcendental but that the GRAPH (which is what I think was meant by "curve") is algebraic. Everyone seems to be agreeing with that.

10^1.5= 10^(3/2)= sqrt(1000). According to the calculator included with Windows, that is 31.6227.... .Actually with the Windows calculator it is easier to do it directly using the "x^y" key. Enter 10, then click "x^y", then enter 1.5 and =. Notice that I did 10^(3/2)= sqrt(1000) by doing 10^3 first, then (1/2) power= square root. You could also do it the other way: take the square root of 10, then the third power. Of course, if you are doing it "by hand", that second way is harder.

You couldn't use this as a proof of that there are an infinite number of primes because you have now assumed the conclusion! I'm really just objecting to your specific wording. I think you have the correct idea.

First, let me point out that the " Because there where the same density they hit the ground at the same time" is not true. The density is irrelevant. Also what Galileo really did is roll object down inclines. That way he could control the acceleration and so speed so he could watch more carefully. Of course, he found that both light and heavy objects rolled down the incline in the same time. As long as you are using objects that are fairly heavy for their size (iron, small wood spheres, etc.) air restistance doesn't mess it up too badly. What he as really trying to do is disprove Aristotle's claim that the acceleration would be in PROPORTION to the mass so small differences in time were not a greate problem.

What are your thoughts on this. What kinds of problems do you have in linear algebra. One of the first that comes to mind is solving systems of equations. I'm not a chemistry major but I have dim memories of trying to "balance" chemical formulas. Doesn't that involve systems of equations? And deciding how much of a particular chemical to add to a mixture seems to involve systems of equations.

Did you try this: http://www.math.com/tables/algebra/conics.htm or this: http://xahlee.org/SpecialPlaneCurves_dir/ConicSections_dir/conicSections.html

As far as your last question, "Lastly, i was playing around with my TI-89, and decided to graph y=sin(x) in 3D. It came up as a plane instead of a line. I guess I'm just having a hard time picturing why that is." is concerned, why would you expect it to be a line? If you were to graph x= 0 in 1D (i.e. a number line) it would be a single point. In 2D, it would be a vertical line (the y-axis), because each point has both x and y coordinates now and "x= 0" allows y to be anything. In 3D, "x= 0" is a plane (the yz-coordinate plane) because both y and z can be anything. Another way of looking at it is this: Selecting an arbitrary point in 3 dimensions you have 3 "degrees of freedom"- you can choose any 3 numbers you want. If you have one equation, f(x,y,z)= 0, say, then you can (theoretically) choose any two numbers for two variables and solve for the third variable. You have two "degrees of freedom" and the graph is a surface. (And, of course, the graph of y= sin(x) is a general surface, NOT a "plane".)

Hey, guys, in the first couple of posts it became clear that he meant "volume" or "density', NOT mass.

I've always though HUMANS were the really invasive species. I think that clip proves it!

Peeing in the wash water? I agree with Cuthber- rinse the cloth very thoroughly! Very, very thoroughly!

Notice that "pulling through the water" is a very different matter from whether something will float or will sink faster than something else. As I said before, the gravitational force is the weight and so will be the same for any two objects that weigh the same will have the same downward force. However, the UPWARD force of bouyancy is equal to the weight of water of the same volume and so is proportional to the volume. The force pulling downward on an object is the difference between the two (if the second is larger, the object floats). However, pulling an object through water has nothing to do with gravity. The force necessary to pull something at constant speed is the "drag" on the object. That tends to be (approximately) proportional to the speed times the cross section area perpendicular to the direction of motion: which for a sphere is pi times r^2. (I feel sure it is cross sectional area, not surface area but I may be wrong.)

You posted a question, got one response that said "first multiply both sides by 10+ r" and just ignore it? Okay, first, do you really mean 13.8= 5 lg (1000r/10)+ r or 13.8= 5 (lg(1000r)/10)+ r or 13.8= 5 lg(1000r)/(10+ r)? insane alien was assuming you meant the last. But this is true only if you mean the first. Assuming you meant the first of the three options above, this is correct so far. No. "2 lg r" is 2 MULTIPLIED by lg r. The "inverse" of that is to DIVIDE by lg r, not subtract it.

I have read that the following took place during WWII: Classes were given on how to set up and use the new (and very secret) "Norden Bombsite". Because of the secrecy, people were not allowed to take any notes outside the classroom- but it was found that the people who took notes anyway and then turned them in at the end of the class did better than people who did not take notes. If you are a good note-taker, taking notes CONCENTRATES your listening rather than interfering with it.

I must confess I've never seen a open a plea before! "Please do my homework for me"! Do you understand that that is pretty much a guarenteed way to fail the course? Teachers give you homework in order to help you learn- often they use homework as feed-back- if people do badly on the homework on a particular topic, they review it. Almost all of your grade (in many cases ALL of it) depends on test grades, not homework. If you have someone else do your homework for you, so you turn in perfect homework despite knowing nothing about it, your teacher will think, "Great, we can go on to new material"- when you take the test and still know nothing about the material, it's too late- you fail the test and then the course.

Perhaps it goes to show that you do not read carefully. The item you refer to specifically says that it is giving a definion of planets IN OUR SOLAR SYSTEM.

The same weight but different mass?? How are you going to arrange that? WEIGHT is defined as "force of gravity acting on an object" and is proportional to mass- you cannot have two objects with the same weight but different mass. Did you mean "same weight but different VOLUME". In that case, the object with less volume would sink faster. Since the two objects have the same weight, the gravitational force downward would be the same for both objects. However, the upward bouyant force is proportional to volume (it is the density of water times the volume) so the object with greater volume will have a larger upward force subtracted from the downward, gravitational, force.

Your quote says "what matters is the point set (typically in the plane) underlying C. Do they give a definition of "algebraic SET" or "transcendental SET"? (edit: Yes, it does: "In mathematics, an algebraic set over a field K is the set of solutions in Kn (n-tuples of elements of K) of a set of simultaneous equations P1(X1, ...,Xn) = 0 P2(X1, ...,Xn) = 0 " That definition, and yours, is in the field of "algebraic varieties" and has nothing to do with the question here.

The FUNCTION, from R to R^2, f(x)= (cos(x),sin(x)) is indeed a "transcendental function". It's GRAPH happens to be the same as the graph of the (algebraic) relation x2+ y2= 1. Many different functions or relations, whether "algebraic", "transcendental" or whatever, may have exactly the same graph.