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If I know that ∭∇f•∇g dV = ∭∇f•∇f dV = ∭∇g•∇g dV over the region D in space, where f and g are functions of x, y, and z, how can I prove that ∇f=∇g over the region D?

I'm trying to extrapolate CO2 parts per million in the atmosphere into the future. According to http://en.wikipedia.org/wiki/List_of_countries_by_... the world produces 31 billion tonnes of CO2 annually. Also, 2.13 billion tonnes of CO2 equal one ppm in the atmosphere, according to http://cdiac.ornl.gov/pns/convert.html.

This calculates to 14.7 ppm per year of CO2 added to the atmosphere.

However, according to http://www.wolframalpha.com/input/?i=carbon+dioxid... CO2 is only rising at about 2 ppm per year.

Is there something wrong with my sources or calculations? Are there better sources I should use instead?

Per dollar spent on wood pellets, how many tons of CO2 are released?

Choice a: 5.00m

Justification: Add the two numbers to get 10.0m. Divide by the counted number two to get 5.00m.

(a+b)/2

Choice b: 5.0m

Justification: The difference between the two numbers is 2.0m. Add half of this difference, 1.0m, to the smaller distance, 4.0m, and get 5.0m.

a+(b-a)/2

So which one should I use, addition or subtraction, and why?

Notice that in choice "a" I'm using the definition of arithmetic mean and the strict rules of significant figures, and I seem to be gaining precision.

I know that correlation dos not imply causation.

I know that modern global climate change is caused by humans.

I watched The Inconvenient Truth recently, and Al Gore seems to be arguing that the correlation of CO2 to Temperature implies causation.

So, my question:

How can correlation be proven to be causation? Do you need a convincing argument, like the greenhouse effect? Do you need a scientific study?

For the expression x^2+kx+12, where k is an integer and k<0, which of the following are possible values of k?

a. -13

b. -12

c. -5

d. 4

e. The answer cannot be determined

This question was in the Princeton review SAT prep book. It says the answer is a. I think the answer is e. And yes, this is directly quoting the question.

Who is right?

I want to find the harmonic mean of (4, 0, 3, 7)

All online calculators I've used (wolfram alpha, etc.) say that it is 0, but using the formula for harmonic mean, I get a divide by zero error, not 0.

I'm using the formula H = n/(1/x1+1/x2 +...1/xn).

What formula should I use to avoid dividing by zero?

It gives a physical advantage, just like training and genetic advantages.

It doesn't harm anyone, except sometimes the user; the user asked for it and should know the risks.

Not everyone has equal access, but the same goes for training and genetic advantages.

So why is it considered unfair?

I'm asking this as an outsider looking into the world of professional sports.

e^(2πi) = 1, so (e^(πi))² = 1. When x²=1, x=±1. Shouldn't e^(πi)=±1? What prevents e^(πi) from equaling 1?

Also,

e^(2πi) = 1, so one of the 2πi roots of 1 must be e. But e is not on the unit circle in a complex plane. How is this possible?