Question

ques

Answer 1

In an orthogonal curvilinear coordinate system (u,v,w,)
We have the scalar factors
\[h_{1}=|\frac{\partial \vec r}{\partial u}| \space ; \space h_{2}=|\frac{\partial \vec r}{\partial v}|\space ; \space h_{3}=|\frac{\partial \vec r}{\partial w}|\]
Are these factors constant or they vary with u,v,w??
Because in case of cartesian coordinate system where we have
\[\vec r=x \hat i+y \hat j+z \hat k\]\[|\frac{\partial \vec r}{\partial x}|=1 \space ; \space |\frac{\partial \vec r}{\partial y}|=1 \space ; \space |\frac{\partial \vec r}{\partial z}|=1\]
\[\vec r=X(x,y,z)\hat i+Y(x,y,z)\hat j+Z(x,y,z) \hat k\]
Are the cartesian coordinates "a special case" where
\[X(x,y,z)=x \space ; \space Y(x,y,z)=y \space ; \space Z(x,y,z)=Z \space ; \space\]
The 3 functions are just in fact single variable and simple
because in some orthgonal curvilinear system we may have
\[\vec r=X(u,v,w)\hat e_{1}+Y(u,v,w)\hat e_{2}+Z(u,v,w) \hat e_{3}\]
Now this can be something like
\[\vec r=uv^2w \hat e_{1}+uvw^3 \hat e_{2}+uvw \hat e_{3}\]

Answer 2

@IrishBoy123

Answer 3

@Empty