# How is it that adding a random field to the partial derivative results in a tensorial operation?

We know that the partial derivative of a tensor is not a tensor. But how is this problem fixed by adding to the partial derivatives, a field of Christoffel symbols? Christoffel symbols are a completely arbitrary field because you can define any arbitrary connection you want. How is it that adding a completely arbitrary field fixes the problem, and that the resultant covariant derivative transforms as a tensor?

• Have a look here Apr 15, 2022 at 7:25

Christoffel symbols are a completely arbitrary field because you can define any arbitrary connection you want.

No, the Christoffel symbols are not arbitrary. They are defined (see Christoffel symbols - Definition in Euclidean space) by how the base vectors $$\mathbf{e}_i$$ depend on the coordinates $$x^j$$. $$\frac{\partial\mathbf{e}_i}{\partial x^j} = \Gamma^k_{ij}\ \mathbf{e}_k$$ or equivalently $$d \mathbf{e}_i = \Gamma^k_{ij}\ \mathbf{e}_k\ dx^j \tag{1}$$

It is this definition, from which you can derive that for a tensor field $$A^i$$ the expressions $$\frac{\partial A^i}{\partial x^j}+\Gamma^i_{jk}\ A^k$$ are components of a tensor, while the partial derivatives $$\frac{\partial A^i}{\partial x^j}$$ are not.

You can derive this in a straight-forward way by starting with the invariant differential $$d\mathbf{A}$$ of a vector field between two positions in space.

\begin{align} d\mathbf{A} &= d(A^i\ \mathbf{e}_i) \\ &= dA^i\ \mathbf{e}_i + A^i\ d\mathbf{e}_i & \text{use definition (1)} \\ &= \frac{\partial A^i}{\partial x^j} dx^j\ \mathbf{e}_i + A^i\ \Gamma^k_{ij}\mathbf{e}_k\ dx^j & \text{in the second term swap indices i and k} \\ &= \frac{\partial A^i}{\partial x^j} dx^j\ \mathbf{e}_i + A^k\ \Gamma^i_{kj}\mathbf{e}_i\ dx^j \\ &= \left( \frac{\partial A^i}{\partial x^j}+A^k\ \Gamma^i_{kj} \right) \mathbf{e}_i\ dx^j \end{align} or equivalently $$\frac{\partial\mathbf{A}}{\partial x^j} = \left( \frac{\partial A^i}{\partial x^j}+A^k\ \Gamma^i_{kj} \right) \mathbf{e}_i$$

You see, the covariant derivative emerged as the components of $$\partial\mathbf{A}/\partial x^j$$ in a quite natural way.

• Christoffel symbols can be choosen to one's wills.. the choice shows about how the space he is working on is curved. And, could you give a source for this idea of the 'invariant' differential, this is the first time I've ever seen it. Apr 15, 2022 at 14:08
• And secondly that wiki section you linked clearly states that the symbols discussed for euclidean space only Apr 15, 2022 at 14:09
• @Buraian I don't remember where I saw this approach long time ago. The closest thing I could find now is video The Covariant Derivative (Component Definition) which is part of a larger video series about Tensor Calculus. Apr 15, 2022 at 15:16