# Surface Constant: What Does It Mean & Impact on Gradient?

• Summer2442
In summary, the conversation discusses the meaning of a surface's equation being equal to a constant and its relationship to the gradient of the surface. It is explained that a surface's equation cannot be a constant, but can be written in the form f(x,y,z)=constant. It is also clarified that the gradient of a function of several variables is normal to the surface at each point. The concept of a tangent plane and normal field to a family of surfaces is also mentioned.
Summer2442
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

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.

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Summer2442 said:
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.
It doesn't really make sense to say that an equation is equal to a constant, but, as others have pointed out, the equation decribing any surface can be written in the form f(x,y,z)= constant. Nor does it make sense to talk about the gradient of a surface. I presume what you mean is the gradient of the function. If you describe a surface as f(x,y,z)= constant, the the gradient of f is a vector normal to the surface at each point. That's discussed in any Calculus class in which the gradient of a function of several variables is defined, isn't it?

I am sorry i am a beginner in calculus, so there is a lot that i lack, yes i guess that's the point of my confusion, because I recall that the gradient is tangent to the function, but i guess this only applies for a function dependent on only one variable. but when the function is of several variables then the gradient of that function is normal to the surface at each point. is that right?

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?

Last edited:
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.

No I can visualise, perfect, thanks a lot that really clarified things.

thanks everyone.

## 1. What is the surface constant and how is it calculated?

The surface constant is a numerical value that represents the average elevation of a surface. It is calculated by dividing the sum of all elevation values by the total number of data points.

## 2. How does the surface constant impact the gradient of a surface?

The surface constant has a direct impact on the gradient of a surface. A higher surface constant means a flatter surface, resulting in a smaller gradient. Conversely, a lower surface constant indicates a steeper surface and a larger gradient.

## 3. What is the significance of the surface constant in topographical mapping?

The surface constant is crucial in topographical mapping as it provides important information about the overall elevation of a surface. It helps in determining the slope of the surface and identifying potential hazards such as steep cliffs or valleys.

## 4. Can the surface constant change over time?

Yes, the surface constant can change over time due to natural processes such as erosion and deposition. Human activities such as construction or mining can also alter the surface constant of an area.

## 5. How is the surface constant used in engineering and construction projects?

The surface constant is used in engineering and construction projects to determine the most suitable location for building structures. It helps in identifying areas with a more stable surface and predicting potential challenges such as erosion or landslides.

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