- #1
etotheipi
Hey everyone, I was trying to learn in an unrigorous way a bit about making derivatives in the general manifold, but I'm getting confused by a few things. Take a vector field ##V \in \mathfrak{X}(M): M \rightarrow TM##, then in some arbitrary basis ##\{ e_{\mu} \}## of ##\mathfrak{X}(M)## we have$$\frac{\partial V}{\partial x^\rho} = \frac{\partial}{\partial x^\rho} \left( V^{\mu} e_{\mu} \right) = \frac{\partial V^{\mu}}{\partial x^\rho} e_{\mu} + V^{\mu} \Gamma^{\sigma}_{\rho \mu} e_{\sigma} = \frac{\partial V^{\mu}}{\partial x^\rho} e_{\mu} + V^{\sigma} \Gamma^{\mu}_{\rho \sigma} e_{\mu} = \left(\frac{\partial V^{\mu}}{\partial x^\rho} + V^{\sigma} \Gamma^{\mu}_{\rho \sigma} \right) e_{\mu}$$because of definition of the symbols ##\partial_a e_b = \Gamma^{c}_{ab} e_c##. Then we define a "covariant derivative" as a (1,1) tensor field$$\nabla V = \left(\frac{\partial V^{\mu}}{\partial x^\rho} + V^{\sigma} \Gamma^{\mu}_{\rho \sigma} \right) e_{\mu} \otimes e^{\rho} \equiv {(\nabla V)^{\mu}}_{\rho} e_{\mu} \otimes e^{\rho}$$This tensor can naturally be seen as a map from the tangent space to itself, e.g. if we act the tensor upon an arbitrary basis vector ##e_{\nu}##$$\nabla V(\, \cdot\,, e_{\nu}) = \left(\frac{\partial V^{\mu}}{\partial x^\nu} + V^{\sigma} \Gamma^{\mu}_{\nu \sigma} \right) e_{\mu} = \frac{\partial V}{\partial x^{\nu}}$$My first question is that sometimes I have seen the notation ##\nabla_{\nu} V## used, but never seen it defined. Is this defined to be ##\nabla_{\nu} V \equiv \nabla_{e_{\nu}} V \equiv \partial_{\nu} V##, as above, or something else?
Secondly, say we have some other vector field ##U \in \mathfrak{X}(M): M \rightarrow TM##, then how do we define ##\nabla_{U} V##, which is a function ##\mathfrak{X}(M) \times \mathfrak{X}(M) \rightarrow \mathfrak{X}(M)##? Maybe something like$$\nabla(U,V) = \nabla(U^{\mu} e_{\mu}, V)\equiv \nabla_{U} V = \nabla_{U^{\mu} e_{\mu}} V = U^{\mu} \nabla_{e_{\mu}}V = U^{\mu} \partial_{\mu} V$$and I was wondering if this is correct? Also, if ##f## is some function and ##W## a vector field, then why does ##\nabla_{fW} = f \nabla_W##?
Thanks!
Secondly, say we have some other vector field ##U \in \mathfrak{X}(M): M \rightarrow TM##, then how do we define ##\nabla_{U} V##, which is a function ##\mathfrak{X}(M) \times \mathfrak{X}(M) \rightarrow \mathfrak{X}(M)##? Maybe something like$$\nabla(U,V) = \nabla(U^{\mu} e_{\mu}, V)\equiv \nabla_{U} V = \nabla_{U^{\mu} e_{\mu}} V = U^{\mu} \nabla_{e_{\mu}}V = U^{\mu} \partial_{\mu} V$$and I was wondering if this is correct? Also, if ##f## is some function and ##W## a vector field, then why does ##\nabla_{fW} = f \nabla_W##?
Thanks!