Show that the point satisfies the conditions

[mathmari]

Hey!

A round membrane in space, is over the space $x^2+y^2 \leq a^2$.

The maximum coordinate $z$ of a point of the membrane is $b$.

We suppose that $(x, y, z)$ is a point of the inclined membrane.

Show that the respective point $(r , \theta , z)$ in cylindrical coordinates satisfies the conditions $$0 \leq r \leq a \ , \ \ 0 \leq \theta \leq 2 \pi \ , \ \ |z| \leq b$$
Could you give me some hints how we could show that??

 

[mathmari]

I have done the following:

Using cylindrical coordinates we get $r=\sqrt{x^2+y^2}=\leq |a|$ and sonve $r \geq 0$ we have $0 \leq r \leq a$.

From the defintion of cylindrincal coordinates we have that $0 \leq \theta \leq 2 \pi$,

But how do we get that $|z| \leq b$??(Wondering) By the sentence "The maximum coordinate $z$ of a point of the membrane is $b$" ?? (Wondering)

 

[I like Serena]

Hey! (Smile)

I have done the following:

Using cylindrical coordinates we get $r=\sqrt{x^2+y^2}=\leq |a|$ and sonve $r \geq 0$ we have $0 \leq r \leq a$.

From the defintion of cylindrincal coordinates we have that $0 \leq \theta \leq 2 \pi$,

Good! :)

But how do we get that $|z| \leq b$??(Wondering) By the sentence "The maximum coordinate $z$ of a point of the membrane is $b$" ?? (Wondering)

The sentence "The maximum coordinate $z$ of a point of the membrane is $b$" means the following.
Suppose we have a point on the membrane with coordinate $z$.
Then the size of that coordinate $z$ has a maximum of $b$.

Let the membrane be given by some function $f$ such that:
$$z=f(x,y)$$
with a domain given by $x^2+y^2\le a$.

Then:
$$\text{The maximum (size of the) coordinate z} = |\max_{x^2+y^2\le a} z| = |\max_{x^2+y^2\le a} f(x,y)| = b$$

 

[mathmari]

The sentence "The maximum coordinate $z$ of a point of the membrane is $b$" means the following.
Suppose we have a point on the membrane with coordinate $z$.
Then the size of that coordinate $z$ has a maximum of $b$.

Let the membrane be given by some function $f$ such that:
$$z=f(x,y)$$
with a domain given by $x^2+y^2\le a$.

Then:
$$\text{The maximum (size of the) coordinate z} = |\max_{x^2+y^2\le a} z| = |\max_{x^2+y^2\le a} f(x,y)| = b$$

I haven't understood why it stands that $|\max_{x^2+y^2\le a} f(x,y)| = b$ and not $\max_{x^2+y^2\le a} f(x,y) = b$. Could you explain it to me ?? (Wondering)

 

[I like Serena]

I haven't understood why it stands that $|\max_{x^2+y^2\le a} f(x,y)| = b$ and not $\max_{x^2+y^2\le a} f(x,y) = b$. Could you explain it to me ?? (Wondering)

It's sloppy usage of the wording "maximum coordinate".
Apparently what is intended is the "maximum size of a coordinate" or the "magnitude of a coordinate" or the "absolute value of a coordinate". (Nerd)