# Contravariant derivative?

• yucheng
In summary, the laplacian is defined as the divergence of the gradient of a function. However, in order to use this formula, the gradient must first be expressed in terms of contravariant components. In Pavel Grinfeld's book, the author defines the contravariant derivative as ##\nabla^{i} V = Z^{ij}\nabla_{j} V##, but this definition is not commonly used and there is some confusion surrounding it. The contravariant derivative is typically defined as ##\nabla_{i} F^{j} = \frac{\partial F^{j}}{\partial Z^{i}} + F^{k} \Gamma^{j}_{ki}##, but it

#### yucheng

Homework Statement
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Relevant Equations
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1. The laplacian is defined such that
$$\vec{\nabla} \cdot \vec{\nabla} V = \nabla_i \nabla^i V = \frac{1}{\sqrt{Z}} \frac{\partial}{\partial Z^{i}} \left(\sqrt{Z} Z^{ij} \frac{\partial V}{\partial Z^{j}}\right)$$

(##Z## is the determinant of the metric tensor, ##Z_i## is a generalized coordinate)

But

$$\vec{\nabla} V = \vec{e}^i\nabla_{i} V = \vec{e}^{i} \partial_i V$$

So we need to express the gradient in terms of contravariant components before using the divergence formula.

In Pavel Grinfeld's book, the author claims that $$\nabla^{i} V = Z^{ij}\nabla_{j} V$$ and the author calls this the 'contravariant' derivative.

However, I am concerned because:
https://math.stackexchange.com/ques...erivative-or-why-are-all-derivatives-covarian

and I cannot find the term 'contravariant derivative' anywhere. So...?

Since the contravariant derivative is defined as

$$\nabla_{i} F^{j} = \frac{\partial F^{j}}{\partial Z^{i}} + F^{k} \Gamma^{j}_{ki}$$ ,is there a similar interpretation for $$\nabla^{i} F^{j}$$?

P.S. This might be of interest, but maybe not

https://math.stackexchange.com/ques...-application-of-contravariant-derivative-on-a

Last edited:
You can raise the index on ##\nabla## with the metric as per usual, i.e. ##\nabla^i = Z^{ij} \nabla_j##

ergospherical said:
You can raise the index on ##\nabla## with the metric as per usual, i.e. ##\nabla^i = Z^{ij} \nabla_j##
But is there an interpretation for

$$Z^{mi} \nabla_{i} F^{j} = Z^{mi}\frac{\partial F^{j}}{\partial Z^{i}} + F^{k}Z^{mi} \Gamma^{j}_{ki}$$?