>>12557910take a point p on your sphere. make a map of the region around your point to the cartesian plane R^2. Now every nearby point on the sphere can be mapped to a point in the plane. This is called a chart. Now consider an arbitrary function f(p) on the sphere (eg: pressure). you can map that function to a function in the plane using your chart. Now consider the partial derivatives with respect to x and y of this function. Actually, just consider the partial derivative itself: d/dx and d/dy. These are what we call tangent vectors. Actually, they are called "chart-induced basis vectors" because x and y are the axes of our chart. Every tangent vector in general can be expressed as a linear combination of d/dx and d/dy: tangent vectors are directional derivatives. Note that if we chose a different chart, we would have a different basis, and therefore the same tangent vector would have different components.
A covector is a linear map that takes a vector and returns a scalar. Recall that d/dx is a vector at point p. Recall that df/dx is the derivative of the function with respect to x at point p. Notice that this returns a scalar. A covector at point p is just a differential df, for some arbitrary function f. The chart-induced basis covectors are dx and dy. Notice how by definition dx/dx=1, dx/dy=0, dy/dx=0, dy/dy=1.
Note that vectors and covectors are objects defined at a point, p. If you had an object that defined a co/vector for the entire manifold at every p, you call that a co/vector field.
A (m, n) tensor at point p is a linear map that takes in m covectors and n vectors, and returns a scalar. If you substitute in basis vectors and basis covectors, then you get the components of the tensor. You can construct tensor fields similarly.
Covectors and vectors have their components transform differently under change of basis. You can use these transformations to derive how tensor components transform under change of basis as well.
