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