Indeed, the fundamental statement of relativity is that physical laws must be expressible in terms of these three types of elements, and indeed I think you'll find, if you do a bit more reading on tensors, that vectors and scalars are actually subclasses of tensors.Īs for finding a source that considers vectors to be coordinate invariant, I can't possibly imagine where I would look for one that doesn't. Vectors, tensors, and scalars all have the property of being coordinate invariant, and that's why special relativity is is built from all three. The transformation rules between the |x> and |x'> and the X and X' must be such that the above equality holds, i.e., that |v> is the same entity regardless of coordinate choice. That's why |v> appears at both ends of the chain I wrote. You will note that |v> is just what it is- it is entirely independent of the choices for |x> or |x'>. Where things inside | > are vectors, things that look like |x> are 'basis vectors' and things that look like X are 'components' (and equal the dot product between |v> and |x>, often written ). I can write the general expression like this:

Your key will be understanding the difference between the vector itself, and its components in some coordinate system.

Pick up any book or website devoted to the use of vectors in physics, you will find that vectors are considered coordinate invariant, and that's why they appear in theories like special relativity. I'm afraid that is not at all the point of a vector. The whole point of a vector is that it expresses a quantity in a particular coordinate system, as a function of that coordinate system. It is not the case that vectors are inherently independent of coordinates. A lot of places try to teach us that a vector is a list of numbers, so the vector is its components, but that is not correct- the vector has a life of its own that transcends the components, that's the value in picturing vectors as arrows instead of lists of components. Note the advantage in thinking in Andrew Mason's terms- the invariance of the dot product does not have to seem like magic if you recognize that the vectors are themselves invariant, so will generate invariant scalars when you dot them.

Those are both true, and both approaches end up with the same work because the dot products come out the same if you regard the two vectors being dotted as being coordinate independent (which they are), or if you regard their components as being coordinate dependent but in such a way as to keep the dot products the same. Andrew Mason is saying that the force and displacements vectors (pictured as arrows if you like) are independent of coordinates, and Steely Dan is saying that the vector components depend on coordinates. I think we have a simple confusion here between the difference between a vector, and its components in some coordinates. We should agree that vectors and scalars are both independent of coordinates (indeed, that is pretty much the reason for using them in physics, their objective character). The force and displacement will always have the same magnitude, and the angle between them will always be the same, but nevertheless the force and displacement are different in different coordinate systems. What you are saying (and what is true) is that the magnitude of the force (which is a scalar quantity) does not change from coordinate system to coordinate system. The vector's components do in fact change depending on the coordinate system. The incremental change of displacement of m from position 1 to position 2 is always the displacement from the origin at 2 MINUS the displacement from the origin at 1.įorce is a vector quantity. This displacement is always determined by the displacement vector of the centre of mass of m from the origin MINUS the displacement of the centre of mass of M from the origin. The displacement of the centre of mass of m from the centre of mass of M determines the force. Neither the force nor displacement depend on the co-ordinate system.

Line integral workdone