# Definition:Equal Surface

## Definition

Let $R$ be a region of space, which may be the interior of a body.

Let there exist a point-function $F$ on $R$ giving rise to a scalar field.

An **equal surface** is a surface $S$ embedded in $R$ at which, for all $P \in S$, $\map F P$ is constant.

## Also known as

An **equal surface** is also known as a **level surface**.

## Examples

### Isothermal Surface

An **isothermal surface** is an equal surface $S$ embedded in a body $B$ with respect to a temperature field within $B$.

That is, it is a surface $S$ embedded in $B$ on which the temperature is equal throughout $S$.

### Equidensity Surface

An **equidensity surface** is an equal surface $S$ embedded in a body $B$ with respect to a density field within $B$.

That is, it is a surface $S$ embedded in $B$ on which the (mass) density is equal throughout $S$.

### Equipotential Surface

An **equipotential surface** is an equal surface $S$ embedded in a region of space $R$ with respect to an electric potential field (or in fact any conservative field) within $B$.

That is, it is a surface $S$ embedded in $R$ at which the electric potential is equal throughout $S$.

## Sources

- 1951: B. Hague:
*An Introduction to Vector Analysis*(5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $5$. Scalar and Vector Fields