I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis i , j , k so by invariance nature of vectors, component of gradient must be in contravariant form. However we know by transformation properties and chin rule we find it is a covariant vector. What is wrong with me ? My second question is : if gradient has been written in covariant form, what is the contravariant form of gradient ? Thank you.
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1The differential $df$ is a covariant object, while the gradient $\nabla f$ is contravariant. In coordinates in fact you have $$df=\frac{\partial f}{\partial x^k}dx^{k}\qquad \nabla{f}=g^{ik}\frac{\partial f}{\partial x^k}\frac{\partial}{\partial x^i}$$ – Dario Jul 16 '14 at 10:03
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Cross-posted from http://physics.stackexchange.com/q/126740/2451 – Qmechanic Jul 31 '14 at 12:01