What is the Proper Way to Take Derivatives of Tensors?

[quasar_4]

Not sure if this is the right place for this but...

I am a bit confused about taking derivatives of tensors. Let's say I have some tensors, R, S and T, and an expression like

$$R^{abc} \nabla_a S_{bcd} T^{d}$$.

Do I contract on the d index inside, to get an expression like

$$R^{abc} \nabla_a U_{bc}$$ where U is a new tensor,

then contract with the R tensor on the outside, e.g.

$$\nabla_a V^{a}$$ where V is yet another tensor? Or, can I not contract at all with things that are on two different sides of a differential operator?

I am also confused as to why suddenly the derivatives of scalars don't vanish... I guess the idea is that a derivative should raise the index of a tensor, and since a scalar is a rank 0 tensor, one index should go to 1. But why? I don't intuitively have a good feel for it (i.e., how does the derivative of a scalar act as a map that takes a vector as its argument?).

 

[Ben Niehoff]

Contraction follows the same rules with respect to differentiation as multiplication does. That's why it's written to resemble multiplication. So you can't do most of what you're describing; it would violate the product rule for the derivative operator.

The gradient of a scalar belongs to the dual vector space because to get the rate of change in a direction $\hat u$ you act on the vector via

$$\nabla f \cdot \hat u = df(\hat u) = u^a \partial_a f$$