Hello all, After a brief break from attempting to learn tensor calculus, I'm once again back at it. Today, I started reading this: http://web.mit.edu/edbert/GR/gr1.pdf. I got to about page 4 before things stopped making sense, right under equation 3. Question 1: apparently a "one-form" is a scalar function of a vector; thus, if P is a one-form and V is a vector, then P(V) makes sense. However, V(P) obviously doesn't; a vector isn't a function. Can someone please explain this? Regardless of my confusion, I continued on. Immediately under the introduction to tensors on page 5, it says that "a tensor of rank (m,n) is defined to be a scalar function of m one-forms and n vectors. Thus, a vector is a tensor of rank (1,0)..." Question 2. How can this be so? Shouldn't a vector be a tensor of rank (0,1), based on the above definition? I ran into my biggest confusion on page 6 under the metric tensor. At the top, it says that the metric tensor is a "symmetric bilinear scalar function of two vectors" (just like before- the tensor was said to be a scalar function). Lower on the same page, it says, "thus, the metric g is a mapping from the space of vectors to the space of one-forms." Question 3. What???? Didn't it just say that tensors map vectors and one-forms to scalars? I'm really sorry if these questions are trivial. I would greatly appreciate if anyone could help me out here, because I'm getting a little frustrated.