Definition:Curvilinear Coordinate Surface
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Definition
Let $\KK$ be a curvilinear coordinate system in $3$-space.
Let $\QQ_1$, $\QQ_2$ and $\QQ_3$ denote the one-parameter families that define $\KK$.
Each of the surfaces which is an element of $\QQ_i$ is known as a curvilinear coordinate surface.
Each curvilinear coordinate surface has its unique $q_i$ which is constant for that surface.
Unit Normal
Let $\QQ_i$ be a family of surfaces contributing towards a curvilinear coordinate system in $3$-space.
Let $q_i$ denote the constant identifying a specific curvilinear coordinate surface within $\QQ_i$.
Let $\mathbf a_i$ be a unit normal to such a curvilinear coordinate surface in the direction of increasing $q_i$.
Then $\mathbf a_i$ is the unit normal associated with $\QQ_i$
Sources
- 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (previous) ... (next): Chapter $2$ Coordinate Systems $2.1$ Curvilinear Coordinates