Contravariant derivative of a tensor field in terms of generalized coordinates?

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
  • #1
yucheng
232
57
Homework Statement
N/A
Relevant Equations
N/A
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 $$\nabla^{i} F^{j}$$?

Thanks in advance!

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

https://math.stackexchange.com/ques...-application-of-contravariant-derivative-on-a
 
Last edited:
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  • #2
You can raise the index on ##\nabla## with the metric as per usual, i.e. ##\nabla^i = Z^{ij} \nabla_j##
 
  • #3
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

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

Related to Contravariant derivative of a tensor field in terms of generalized coordinates?

What is a contravariant derivative?

A contravariant derivative is a mathematical concept used in differential geometry to describe how a vector field changes as it moves along a given path. It is a generalization of the concept of a derivative, which is used to describe the rate of change of a function at a specific point.

How is a contravariant derivative different from a covariant derivative?

A contravariant derivative is defined by how a vector field changes along a given path, while a covariant derivative is defined by how a tensor field changes along a given path. In other words, the contravariant derivative is used to describe the change of a vector field with respect to a change in the coordinates, while the covariant derivative is used to describe the change of a tensor field with respect to a change in the coordinates.

What is the notation used for a contravariant derivative?

The notation used for a contravariant derivative depends on the context in which it is being used. In differential geometry, the contravariant derivative is often represented by the symbol ∇, while in physics, it is often represented by the symbol D.

What are some real-world applications of the contravariant derivative?

The contravariant derivative has many applications in physics and engineering, particularly in fields such as fluid dynamics, electromagnetism, and general relativity. It is used to describe the behavior of vector fields in curved spaces, and is essential in understanding the dynamics of particles and fields in these contexts.

How can I calculate a contravariant derivative?

The calculation of a contravariant derivative depends on the specific context in which it is being used. In general, it involves taking the derivative of a vector field with respect to a set of coordinates, and then applying a correction term to account for the change in the coordinates themselves. This can be done using various mathematical techniques, such as the chain rule or the Christoffel symbols.

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