## Surface = constant

Hello,

I wanted to ask what it meant if a surface's equation is equal to a constant. and what does that say about the gradient(surface).

Thanks.
 A surface's equation can't be a constant. A constant is an expression, not an equation.
 So you mean like f(x,y,z)=constant f(x,y,z) = x + y + z = 35 describes a plane in 3 space vs g(x,y,z) = x^2 + y^2 + z^2 = constant describes a sphere in 3 space what can you say about the gradients of these two examples?

## Surface = constant

yes what i mean is f(x, y, z) = constant,

well for f(x,y,z) the gradient is 1xˆ+1yˆ+zˆ (right?), however for g(x, y, z) it is still a function 2xxˆ+2yyˆ+2zzˆ (right?), does that mean it does not matter, that f(x,y,z) = constant has no effect on the gradient but only the function itself.

if that is the case, then why does it say sometimes that when if f(x,y,z) = constant, then the gradient defines the normal to the surface and that it is perpendicular to dr.

thanks.
 If we can describe any surface as f(x,y,z)=g(x,y,z) (which we can for most surfaces, regardless of if they can be represented in terms of elementary functions.) This is equivalent to (f-g)(x,y,z)=0, so we can't really say anything about the surface. I wanted to take the gradient of both sides, but that appears to be invalid.
 let me be more specific :) i hope you can see my attachment, i do not understand the relation between a function f(x, y, z) being equal to a constant, and the gradient of that function being perpendicular to a single increment dr? is there are relation, does f(x, y, z) = constant imply anything. thanks. Attached Thumbnails

Recognitions:
Gold Member
 A general surface is curved and at any point on the surface, X(x1,,y1,z1), there is not a single tangent line, but an infinity of lines forming a tangent plane. The gradient of a function f(x,y,z) = ${\nabla _{{X_1}}}f$ is perpendicular to this plane. Do you need a diagram or can you visualise this?
 Recognitions: Gold Member Science Advisor Hi Summer2442! Consider the family of surfaces $(S_{t})$ which are levels sets of the smooth map $F:\mathbb{R}^{3} \rightarrow \mathbb{R}$. So take an element of the family $S_{t_{0}}$ and any $p\in S_{t_{0}}$ and any $v$ tangent to the surface at that point. Consider a regular curve $c:(-\varepsilon ,\varepsilon )\rightarrow S_{t_{0}}$ such that $c(0) = p, \dot{c}(0) = v$. We have that $F(c(t)) = const, \forall t\in (-\varepsilon ,\varepsilon )$ so $\frac{\mathrm{d} }{\mathrm{d} t}|_{t = 0}F(c(t)) = \triangledown F|_{p}\cdot v = 0$. Since this was for an arbitrary point and tangent vector, we can say the gradient of this smooth map is a normal field to the aforementioned family of surfaces.