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I'm working through Wald's "General Relativity" right now. My questions are actually about the math, but I figure that a few of you that frequent this part of the forums may have read this book and so will be in a good position to answer my questions. I have two questions:

1) Wald first defines a derivative operator ##\nabla## which maps smooth tensors of type ##(k,l)## to smooth tensors of type ##(k, l + 1)##. He defines the operator by its properties. The one that is bothering me is the fourth one. He writes "Consistency with the notion of tangent vectors as directional derivatives on scalar fields: For all ##f \in C^{\infty} M## and all ##t^a \in V_p##, $$t(f) = t^a \nabla_a f$$ What does the notation ##t^a \nabla_a f## mean? It looks like a contraction to me, but Wald explicitly says that the index attached to ##\nabla## is just for notational convenience. I was thinking, if ##t## was a vector field on ##M## (the smooth manifold in question), then ##tf \in C^{\infty} M## and so ##t^a \nabla_ f## could be interpreted as ##\nabla_a tf##, which would make sense. But ##t## is merely a tangent vector, so in this case I am confused.

2) My second question is this: After Wald lists the five properties that the covariant derivative operator is required to satisfy, he shows that such an operator does indeed exist. He says let ##\psi## be a smooth coordinate map on ##M## and let ##\{\partial/\partial x^\mu\}## and ##\{dx^\mu\}## be bases for the tangent space and cotangent space, respectively. Then given a smooth tensor field ##T^{a_1 \cdots a_k}_{b_1 \cdots b_l}##, take its components ##T^{\mu_1 \cdots \mu_k}_{\nu_1 \cdots \nu_l}## in the given coordinate basis and define ##\partial_c T^{a_1 \cdots a_k}_{b_1 \cdots b_l}## to be the tensor whose components are the partial derivatives ##\partial(T^{\mu_1 \cdots \mu_k}_{\nu_1 \cdots nu_l} )/ \partial x^{\sigma}##.

I understand that because ##T## is a smooth tensor field then its components are smooth real-valued maps, and consequently, we can take their partial derivatives. But to which variable do we differentiate with respect to? The components of ##\partial_c T## (which is a type ##(k, l+1)## tensor) are supposed to be the partial derivatives of the component functions of ##T##, but if you ask me to take the partial derivative of a component function of ##T##, I ask, which of the ##n## variables do I differentiate with respect to?

I hope my misunderstandings are clear. If they aren't, please let me know and I will clear it up.

EDIT: In regards to my first question, I just realized that ##\nabla f## is dual to ##df##, and ##t^a## is a vector, and by the isomorphism ##V_p \simeq V_p^{**}##, the vector ##t^a## acts on the dual vector ##df##. So, though ##t^a \nabla_a f## isn't a contraction, writing it as one is indeed notationally convenient because it's a quick way to say that ##t^a## is acting on ##df##. Is this correct?

EDIT 2: I am going to take a quick stab at my second question by trying to answer it with an example. Please let me know if it is correct. Let us take a simple tensor field ##T## of type ##(2,2)## on a smooth ##3##-manifold. Suppose ##T## has the simple expansion, ##T = f \Big( \partial/\partial x^1 \otimes \partial/\partial x^2 \otimes dx^1 \otimes dx^2 \Big)## where ##f## is a smooth real valued function on ##M##. Then ##\nabla T = \frac{\partial f}{\partial x^1} \Big(\partial/\partial x^1 \otimes \partial/\partial x^2 \otimes dx^1 \otimes dx^2 \otimes dx^1 \Big)+ \frac{\partial f}{\partial x^2} \Big( \partial/\partial x^1 \otimes \partial/\partial x^2 \otimes dx^1 \otimes dx^2 \otimes dx^2 \Big) + \frac{\partial f}{\partial x^3} \Big( \partial/\partial x^1 \otimes \partial/\partial x^2 \otimes dx^1 \otimes dx^2 \otimes dx^3 \Big)##. Is this what Wald means?

1) Wald first defines a derivative operator ##\nabla## which maps smooth tensors of type ##(k,l)## to smooth tensors of type ##(k, l + 1)##. He defines the operator by its properties. The one that is bothering me is the fourth one. He writes "Consistency with the notion of tangent vectors as directional derivatives on scalar fields: For all ##f \in C^{\infty} M## and all ##t^a \in V_p##, $$t(f) = t^a \nabla_a f$$ What does the notation ##t^a \nabla_a f## mean? It looks like a contraction to me, but Wald explicitly says that the index attached to ##\nabla## is just for notational convenience. I was thinking, if ##t## was a vector field on ##M## (the smooth manifold in question), then ##tf \in C^{\infty} M## and so ##t^a \nabla_ f## could be interpreted as ##\nabla_a tf##, which would make sense. But ##t## is merely a tangent vector, so in this case I am confused.

2) My second question is this: After Wald lists the five properties that the covariant derivative operator is required to satisfy, he shows that such an operator does indeed exist. He says let ##\psi## be a smooth coordinate map on ##M## and let ##\{\partial/\partial x^\mu\}## and ##\{dx^\mu\}## be bases for the tangent space and cotangent space, respectively. Then given a smooth tensor field ##T^{a_1 \cdots a_k}_{b_1 \cdots b_l}##, take its components ##T^{\mu_1 \cdots \mu_k}_{\nu_1 \cdots \nu_l}## in the given coordinate basis and define ##\partial_c T^{a_1 \cdots a_k}_{b_1 \cdots b_l}## to be the tensor whose components are the partial derivatives ##\partial(T^{\mu_1 \cdots \mu_k}_{\nu_1 \cdots nu_l} )/ \partial x^{\sigma}##.

I understand that because ##T## is a smooth tensor field then its components are smooth real-valued maps, and consequently, we can take their partial derivatives. But to which variable do we differentiate with respect to? The components of ##\partial_c T## (which is a type ##(k, l+1)## tensor) are supposed to be the partial derivatives of the component functions of ##T##, but if you ask me to take the partial derivative of a component function of ##T##, I ask, which of the ##n## variables do I differentiate with respect to?

I hope my misunderstandings are clear. If they aren't, please let me know and I will clear it up.

EDIT: In regards to my first question, I just realized that ##\nabla f## is dual to ##df##, and ##t^a## is a vector, and by the isomorphism ##V_p \simeq V_p^{**}##, the vector ##t^a## acts on the dual vector ##df##. So, though ##t^a \nabla_a f## isn't a contraction, writing it as one is indeed notationally convenient because it's a quick way to say that ##t^a## is acting on ##df##. Is this correct?

EDIT 2: I am going to take a quick stab at my second question by trying to answer it with an example. Please let me know if it is correct. Let us take a simple tensor field ##T## of type ##(2,2)## on a smooth ##3##-manifold. Suppose ##T## has the simple expansion, ##T = f \Big( \partial/\partial x^1 \otimes \partial/\partial x^2 \otimes dx^1 \otimes dx^2 \Big)## where ##f## is a smooth real valued function on ##M##. Then ##\nabla T = \frac{\partial f}{\partial x^1} \Big(\partial/\partial x^1 \otimes \partial/\partial x^2 \otimes dx^1 \otimes dx^2 \otimes dx^1 \Big)+ \frac{\partial f}{\partial x^2} \Big( \partial/\partial x^1 \otimes \partial/\partial x^2 \otimes dx^1 \otimes dx^2 \otimes dx^2 \Big) + \frac{\partial f}{\partial x^3} \Big( \partial/\partial x^1 \otimes \partial/\partial x^2 \otimes dx^1 \otimes dx^2 \otimes dx^3 \Big)##. Is this what Wald means?

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