Tensor field without defining manifolds I'm currently learning tensors. I've learnt scalar and vector fields in 2-D or 3-D. But when I looked at the definition of tensor field, they usually mentioned associating a point on a manifold with a tensor. I kind of know how this works out. But can we discuss the property of tensor fields only on 2-D or 3-D points? Will their property still be maintained?
I'm also curious about how to take derivative of tensor fields, and how tensor fields are usually applied in differential geometry. Thanks!
 A: A tensor is a multilinear function on a vector space $V$. A tensor field is defined as follows: Start with a smooth parametrized family of vector spaces. A tensor field is simply a smoothly varying assignment to each parameter a tensor on the corresponding vector space. The parameter space can be anything you want. The most common setting in differential geometry is the tangent bundle, where the parameter space is comprised of points in a manifold and the vector space associated to each point is the tangent space at that point. So a tensor field would be a tensor on each tangent space in the manifold that depends smoothly on the point in the manifold.
If you use the vector space $V$ itself as the manifold, then a tensor field is simply a smooth map from each vector in $V$ to a tensor on $V$. If you fix a basis of $V$, then you get a basis of the corresponding tensor space and therefore, the tensor field can be written as an array of scalar functions on $V$. You can then differentiate the tensor field with respect to the coordinates on $V$. This kind of differentiation is called a "flat connection".
If $V$ is either $\mathbb{R}^2$ or $\mathbb{R}^3$, then you get the 2D and 3D tensor fields you're mentioning.
If the parameter space is a non-flat manifold, such as a sphere, then differentiation of a tensor field becomes a bit more involved, requiring the concept of what's known as a connection or a covariant derivative.
