Vector projection in polar cordinatesIn Euclidean Space and Cartesian coordinates system, we know the vector projection of Vector u onto a vector v is simply P → = ( u → . v → ) v → v → . v → What would be the vector projection on polar or spherical coordinates or alternate coordinate systems?Suppose we have vectors defined as u → = u α e α v → = v β e β Does a projection vector of u on v becomes P → = ( u α v β e α β ) v → v α v β e α β Where e α represent the basis vector and e α β the metric tensor.

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Vector projection in polar cordinatesIn Euclidean Space and Cartesian coordinates system, we know the vector projection of Vector u onto a vector v is simply            P      →        =            (                        u          →                    .                        v          →                    )                               v          →                                              v          →                    .                        v          →                    What would be the vector projection on polar or spherical coordinates or alternate coordinate systems?Suppose we have vectors defined as            u      →        =      u          α            e          α                  v      →        =      v          β            e          β      Does a projection vector of u on v becomes            P      →        =            (              u                  α                            v                  β                            e                              α                                β                              )                        v          →                                    v                  α                            v                  β                            e                              α                                β                              Where       e          α       represent the basis vector and       e                  α                    β             the metric tensor.

William Collins · Accepted Answer

That looks correct. The dot product formula for projection doesn&#039;t depend on the coordinate system, though you need to know how the dot product actually works, which can be determined by its action on a set of basis vectors.            u      →        ⋅            v      →        =            (            ∑    α        u    α                      e        →              α              )        ⋅            (            ∑    β        v    β                      e        →              β              )        =      ∑          α      ,      β            u    α        v    β    (                    e        →              α    ⋅                    e        →              β    )  =      ∑          α      ,      β            u    α        v    β    (      e          α      β        )But the vectors             u      →        and             v      →        need to be in the same tangent space (based at the same point) in general. And the vectors are not points like   (  r  ,  θ  ) in polar coordinates.

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