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I know that gradient is orthogonal to the tangent plane of any point in the level set,but is it true claiming that it is orthogonal to the level set?I think they are different in fact.

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    What does "orthogonal to the level set" mean, otherwise?2012-03-02
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    i am not sure,look at http://en.wikipedia.org/wiki/Gradient the part about the level set and gradient and you will see2012-03-02
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    Dylan's point is that "orthogonal to the level set" _means_ "orthogonal to the tangent plane of the the level set". Claiming that one of these could be true without the other one also being true is self-contradictory. If you want to convince anyone that, no, they actually mean different things, the onus is on you to explain which different things you want them to mean.2012-03-02

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