Equal Surfaces do not Intersect

Theorem

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.

Let $S_1$ and $S_2$ be equal surfaces in $R$ upon which the value of $F$ on $S_1$ is different from the value of $F$ on $S_2$.

Then $S_1$ and $S_2$ do not intersect.

Proof

Let:

$\forall p \in S_1: \map F p = C_1$

$\forall p \in S_2: \map F p = C_2$

By hypothesis, $C_1 \ne C_2$.

Aiming for a contradiction, suppose there exists a point $P$ in $R$ such that both $P \in S_1$ and $P \in S_2$.

Then $\map F p = C_1$ and also $\map F p = C_2$.

This contradicts the fact that $F$ is a function.

Hence the result by Proof by Contradiction.

$\blacksquare$

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