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Everyone knows Temperature gradient is a vector quantity having direction from cold to hot.My confusion: why is temperature gradient vector if its direction is always fixed (as in the case of pressure) (Don't say that it is because it follows vector law of addition,I am searching for more concrete answer...)

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Temperature gradient is actually an object called a one-form. A temperature gradient does not have a direction. Instead you combine it with a vector to get a scalar (the temperature change). It's the vector that gives the direction.

To take a simple 1-D example, suppose we have a temperature that varies along the $x$ axis as:

$$ T = 298 + x $$

so at $x = 0$ the temperature is 298K, at x = 1m it's 299K and so on. The temperature gradient is obviously 1 degree per metre. If we take a unit vector pointing to the right i.e. $\vec{x} = (1)$ and multiply this by the temperature gradient we get +1 i.e. the temperature rises by 1K when moving 1m in the positive $x$ direction. But suppose we take a unit vector pointing to the left, $\vec{x} = (-1)$ and multiply this by the temperature gradient we get -1 i.e. the temperature falls by 1K when moving 1m in the negative $x$ direction.

So you can't say that the temperature gradient has a direction. It's the vector you choose to multiply it by that gives a direction i.e. hotter in one direction and cooler in the opposite direction.

The example is rather trivial in 1D, but gets more interesting in 3D, and more interesting still in curved co-ordinate systems as found in general relativity. In fact a gradient is the archetypal example of a one-form used in many GR textbooks.

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That's not quite true. One can define the gradient as the result of the metric-generated canonical isomorphism between cotangent and tangent bundles applied to the exterior derivative of a function (the one-form you describe), which provides a direction that's no longer arbitrary. This is a vector that's consistent with the elementary calculus notion of gradient, and so I think preferable to simply using it as a synonym for 'exterior derivative of a function', as you have here. –  Stan Liou Apr 22 '13 at 13:06
    
@StanLiou: your comment tells me I don't understand this subject as well as I thought. I'll leave my answer in place for now, but I will delete if you think it is misleading. –  John Rennie Apr 22 '13 at 19:37
    
@StanLiou Nice ans dude, I checked a lot of book to catch it and was satisfactory.. John, Don't delete it.. It is not misleading.. I voted it up because it was nice.. –  newera Apr 23 '13 at 16:37
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A maybe more mathematical awnser: You can define temperature as a scalar field (e.g. on earth). So given a certain position on the surface of the earth (or in three dimensions if you wish, it does not change anything) you have a scalar, the temperature on this position. Now you can take the gradient of this field, and now you have a vector.

More directly on your question: 1) A vector is still a vector, even if he has a constant value. 2) Why is its value fixed? You don't know where the temperature is highest, or you can even define a time dependend temperature field, (on earth temperature is not fix, it changes quite obviously), then its gradient is not fixed either

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First of all thank you...,secondly what is the meaning of the direction which is always certain, the nearest example of which I previously mentioned. I don't know why you introduced the concept of the dimension and space here well in fact that was not necessary.The real concept that "the gradient of a scalar field is a vector" would have been more than sufficient. At last but not at least I never said vector is not a vector if it has a constant direction, I wanted to ask What is the meaning of the direction if it is always the same no matter whatever happens.. –  newera Apr 23 '13 at 16:22
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