Cartesian to curvilinear coordinate transformations

[Stendhal]

Homework Statement

Is there a more intuitive way of thinking or calculating the transformation between coordinates of a field or any given vector?

The E&M book I'm using right now likes to use the vector field

$\vec F\ = \frac {\vec x} {r^3}$

where r is the magnitude of $\vec x$In Cartesian coordinates, this looks like

$\frac {x \hat x + y \hat y + z \hat z} {\sqrt {x^2 + y^2 +z^2}^3}$

In problems such as finding the flux through a sphere, it's difficult to use cartesian coordinates as it's very algrebra intensive, but I find it hard to convert between different coordinate systems. It also seems really unnecessary to simply look up the values of x,y,z and their respective $\hat x$ directions for the components. Is there a better way to go about thinking and converting fields?

 

[jambaugh]

One of the advantages of using vector notation is that it should be independent of your coordinate system so to answer your question about "a more intuitive way..." I would say; Yes, use the vector notation as much as possible. Where you can, rewrite component formulas in terms of dot products or vector magnitudes. For example if you need the $x$-component of a vector $\vec{v}$ write that as $v_x=\hat{\imath} \bullet \vec{v}$. Granted you'll eventually need to resolve coordinates for such things as integrating over a surface but if you do most of your conceptual work in the general notation first, you often can select the coordinate system that makes this easiest.