I wouldn’t use the phrase “transforms trivially” here since a “trivial transformation” usually refers to the identity transformation
No, I do mean the identity transformation. Scalar fields do not transform at all under coordinate changes. To be precise, if we have a coordinate change matrix P, a scalar field f(x) transforms like f′(x)=f(Px)
Whereas a vector field v(x) transforms like
v′(x)=P−1v(Px)
Ah. Thank you, that is perfectly clear. The Wikipedia page for Scalar Field makes sense with that too. A scalar field is a function that takes values in some canonical units, and so it transforms only on the right of f under a perspective shift. A vector field (effectively) takes values both on and in the same space, and so it transforms both on the left and right of v under a perspective shift.
No, I do mean the identity transformation. Scalar fields do not transform at all under coordinate changes. To be precise, if we have a coordinate change matrix P, a scalar field f(x) transforms like f′(x)=f(Px)
Whereas a vector field v(x) transforms like v′(x)=P−1v(Px)
For more details check out these wikipedia pages.
Ah. Thank you, that is perfectly clear. The Wikipedia page for Scalar Field makes sense with that too. A scalar field is a function that takes values in some canonical units, and so it transforms only on the right of f under a perspective shift. A vector field (effectively) takes values both on and in the same space, and so it transforms both on the left and right of v under a perspective shift.
I updated my first reply to point to yours.