# Conversion between vector components in different coordinate systems

Karl Karlsson
Homework Statement:
I am not completely sure what the formulas ##v_j = v^a\frac {\partial x^j} {\partial \chi^a}## and ##v^b = v^a\frac {\partial \chi^b} {\partial x^j}## mean. Is ##v_j## the j:th cartesian component of the vector ##\vec v## or could it hold for other bases as well? What does the second equation ##v^b = v^a\frac {\partial \chi^b} {\partial x^j}## mean? Is this just the relation between the b:th and j:th component of ##\vec v## being expressed as tangent vectors?
Relevant Equations:
##v_j = v^a\frac {\partial x^j} {\partial \chi^a}## and ##v^b = v^a\frac {\partial \chi^b} {\partial x^j}##
I am not completely sure what the formulas ##v_j = v^a\frac {\partial x^j} {\partial \chi^a}## and ##v^b = v^a\frac {\partial \chi^b} {\partial x^j}## mean. Is ##v_j## the j:th cartesian component of the vector ##\vec v## or could it hold for other bases as well? What does the second equation ##v^b = v^a\frac {\partial \chi^b} {\partial x^j}## mean? Is this just the relation between the b:th and j:th component of ##\vec v## being expressed as tangent vectors?

Staff Emeritus
Hmm. The "relevant equations" seem slightly wrong, to me. I think it should be:

##v^j = v^a \frac{\partial x^j}{\partial \chi^a}## and ##v^a = v^j \frac{\partial \chi^a}{\partial x^j}##

Having said that, the equations are good for translating between any two coordinate systems, whether Cartesian, or not.

Staff Emeritus