Dimension, graph of functions of several variable and it's visualization.

Question

To visualise a scalar function of $n$ variables we consider its graph in $(n + 1)$ dimensional space.

If $\mathit{f} :U \subset \mathbb{R}^n \to \mathbb{R}$ is a function of n variable its graph consists of the set of points $(x_1,....,x_n,\mathit{f}(x_1,....,x_n))$ in $\mathbb{R}^{n+1}$ for $(x_1,....,x_n)$ in $U$

This lines are from my reference.

Considering the above definition.

I suppose, dimension of a set as the number of independent variables (i.e scalars) that is required to describe the position of a point of the set.(also considering for sets which are not vector spaces )

Now, $$\mathrm{S} =\lbrace (x,y,z)\in \mathbb{R}^3 | x^2+y^2+z^2=1\rbrace$$ (this set is not a vector space),now any point of this set can be represented using two scalars $(x,y,\pm \sqrt{1-x^2-y^2})$ So dimension of $\mathrm{S}=2$

$$\mathrm{T} =\lbrace (x,y,z)\in \mathbb{R}^3 | x^2+y^2+z^2\le1\rbrace$$ (also this set is not a vector space) Clearly, the points of this set cannot be represented by $x$ and $y$ alone so we need 3 scalar $(x,y,z)$ to represent the element of this set $\mathrm{T}$. So, dimension of $\mathrm{T}=3$

Physically, $\mathrm{S}$ is a sphere( or better to say ,here a spherical shell ) And $\mathrm{T}$ is a spherical ball.

So, from my perspective of definition of dimension. $\mathrm{S}$ is a $2D$ object and $\mathrm{T}$ is a $3D$ object. Now if we plot this in any 3D grapher we certainly cannot distinguish between this two objects(graphs)

My questions are:

1. Is my view for dimension correct? Is there anything like dimension for non vector space set?

2. Are the sets that represent paraboloid, hyperboloid,or a right circular cylinder all 2D objects?

3. What is the difference between object and its graph?

4. Considering the definition of my book do we need 4D space to visualise the set $\mathrm{T}$? And how it is different in visualization with the set $\mathrm{S}$

I am not sure of the tags i used.

Answer 1

1. Your view of dimension is correct. Indeed, the very first sentence on Wikipedia is what you say:

https://en.wikipedia.org/wiki/Dimension

There are more technical ways of defining it, but your intuition is correct.

2. Yes, hyperboloid, paraboloid, cylinder are all 2-dimensional.

3. I'm not sure what you mean by an "object".

4. No, $T$ can be visualized in 3d space. It is the set of points inside the sphere. It is different from $S$ in that $S$ is only the points on the boundary of the sphere, but $T$ are all the points inside.

The major difference between $T$ and $S$ is that $S$ is defined by an equation, but $T$ is defined by an inequality. Generically, each equation needed to define a surface/manifold decreases the dimension by 1. For example:

• One equation in $(x,y)$ defines a curve (1-dimensional)
• One equation in $(x,y,z)$ defines a surface (2-dimensional)
• Two equations in $(x,y,z)$ defines a curve (1-dimensional)

On the other hand, an inequality does not generically decrease the dimension, as you can see with your $T$ example.