>>11053836a tensor is a function.
a function is a mathematical object, which operates on other mathematical object.
>>11053846when we say "transforms" we mean that when you change your coordinate system, i.e. when you change your perspective, the entries of the tensor change in a certain way.
like how if you were thinking of the projection on the x axis and you were facing the x axis, the tensor would project in front of you, but if you then turned to face the y-axis now the tensor projects to your side.
"transforming like a tensor" means transforming in a particular way when you change perspective.
"changing perspective" is a way to act on a tensor, yes. it is not the same as how the tensor acts on other things.
>>11053873>>11053890a 3 dimensional 3-tensor would have 27 elements, yes. we'd probably call them coordinates.
you're confusing yourself a little i think, an n-tensor takes n vectors. the vectors can have whatever dimension you want them to have. if you're talking about an n tensor which takes m dimensional vectors, then you need m^n coordinates to define the tensor. you can prove this by induction:
for n = 1, you just act on a single m dimensional vector. you need to choose how to act on each of the m components, which takes m numbers.
now suppose there are m^(n-1) coordinates for an (n-1)-tensor. then we add an nth vector. for each possible coordinate of the nth vector, we need m^(n-1) coordinates to tell you how to act on the others (by the inductive hypothesis). then, we need to tell you how to act on the nth vector - well, there are m coordinates for you to pick actions on. so there are m^(n-1)m = m^n coordinates for the n-tensor.
just remember that the dot product is a 2-tensor no matter how long your vectors are, since it always takes 2 vectors.
