Equal Surfaces do not Intersect
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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