1. ## Differential Formulas

I have a question that says:

Write a differential formula of a sphere when the radius of a balloon goes from r0 to r0+ dr.

Can you walk me through this?

2. ## Re: Differential Formulas

For a cube:
The volume of a cube of edge $\displaystyle a$ is $\displaystyle V=a^3$

Thus the change in volume $\displaystyle \mathrm dV$ as the edge goes from $\displaystyle a$ to $\displaystyle a + \mathrm da$ is given by:

$\displaystyle \mathrm dV = (a + \mathrm da)^3 - a^3$

Expanding and simplifying we get

$\displaystyle \mathrm dV = 3a^2\mathrm da + 3a(\mathrm da)^2 + (\mathrm da)^3$

Ignoring non-linear terms of differentials (squares, cubes, etc.) we end up with

$\displaystyle \mathrm dV = 3a^2\mathrm da$

3. ## Re: Differential Formulas

Ok, so for a sphere. The formula is V = 4/3pia^3 . Would i set it up like this:

dV = 4/3pi(a+da)3 - 4/3pi(a)3

4. ## Re: Differential Formulas

Wait. Is this question asking for dy? I think I might just be confused over the wording.

I know dy = f`(a)dx

Is that what is happening when they ask for a differential?

5. ## Re: Differential Formulas

Yeah, that'd be it.

6. ## Re: Differential Formulas

When the radius of the sphere "goes from r to r+ dr" the volume goes from $\displaystyle \frac{4}{3}\pi r^3$ to $\displaystyle \frac{4}{3}\pi (r+ dr)^3= \frac{4}{3}\pi(r^3+ 3r^2dr+3rdr^2+ dr^3)$. The difference is $\displaystyle \Delta y= \frac{4}{3}\pi(3r^2 dr+ 3r dr^2+ dr^3)$. If we take dr sufficiently small that we can ignore the much smaller $\displaystyle dr^2$ and $\displaystyle dr^3$ that would be the differential $\displaystyle dy= 3r^2 dr$.

7. ## Re: Differential Formulas

Originally Posted by HallsofIvy
When the radius of the sphere "goes from r to r+ dr" the volume goes from $\displaystyle \frac{4}{3}\pi r^3$ to $\displaystyle \frac{4}{3}\pi (r+ dr)^3= \frac{4}{3}\pi(r^3+ 3r^2dr+3rdr^2+ dr^3)$. The difference is $\displaystyle \Delta y= \frac{4}{3}\pi(3r^2 dr+ 3r dr^2+ dr^3)$. If we take dr sufficiently small that we can ignore the much smaller $\displaystyle dr^2$ and $\displaystyle dr^3$ that would be the differential $\displaystyle dy= 3r^2 dr$.
Well said, but for clarity, I just want to correct the small typo at the end. $\displaystyle dy = 4\pi r^2 dr$.

8. ## Re: Differential Formulas

Yes, I accidently left out the "$\displaystyle \pi$". Thanks for the correction.