Real Function of Two Variables represents Surface in Cartesian 3-Space
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Theorem
Let $S$ and $T$ be subsets of the set of real numbers $\R$.
Let $f: S \times T \to \R$ be a real function of two variables.
Then the locus of $f$ describes a surface embedded in the Cartesian space $\R^3$.
Proof
This theorem requires a proof. In particular: The book says "clearly represents", but there is a concern that $f$ may need to be defined as continuous. And while it's intuitively obvious, it may not be so easy to prove rigorously. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 1$. Introduction