# Scalar Function on a Surface

## Main Question or Discussion Point

Hi guys and gals

This is a conceptual question. Lets say I have a scalar function, $$f(x,y,z)$$ defined throughout $$\mathbb{R}^3$$. Further I have some bounded surface, S embedded in $$\mathbb{R}^3$$.

How would I find the function f, defined on the surface S?

Would it be the inner product of f and S, $$<f|S>$$ or a functional composition like $$f \circ S$$?

mathman
f is a scalar, so inner product of S and f makes no sense. I don't know what you have in mind by functional composition

you mean you want parameterize f by s? as in restrict f to s? like for the purposes of a surface integral?

f is a scalar, so inner product of S and f makes no sense.
From what I understand the inner product <f|g> is
$$\int_{-\infty}^{\infty}f(t)g^{*}(t)dt$$.

The mistake I made was to think that they are scalar functions aswell even though f and g are complex functions. Sorry about that. The closest thing I've come to inner products for functions was the orthonormality of the basis functions for Fourier series. ice109 said:
you mean you want parameterize f by s? as in restrict f to s? like for the purposes of a surface integral?
This is exactly what I had in mind. Sorry I should have been more explicit where I was going with it.

I understand what we are doing if we have a vector field $$\textbf{F}$$ and want to find out how it permeates (eg. flux through a surface) a surface S but dotting it with the unit normal of the surface and integrating on the surface. This is actually what made me think of the inner product:
$$\iint\textbf{F}\cdot\textbf{n}\,\mathrm{dS}$$

Thanks for the replies HallsofIvy