Derivative of area =circumference

In summary, the derivative of the formula for area of a circle is the formula for circumference, while the derivative of the formula for volume of a sphere is the formula for surface area. These relationships can be explained using Stokes's and Divergence theorems, and can also be understood through integration exercises.
  • #1
HyperbolicMan
14
0
For a circle, the derivative of the formula for area is the formula for circumference:

d/dr([pi*r^2)=2*pi*r.

Similarly, the derivative of the formula for the volume of a sphere is equal to the formula for surface area: d/dr(4/3*pi*r^3)=4*pi*r^2.

I'm positive that these are consequences of Stokes's (Green's) and Divergence theorems, but I don't know how to show this. Could someone please explain? Thanks.
 
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  • #2
Looking at it as an integration exercise makes things clearer. Add up circular lamina of length 2πr and width dr to get area. Similarly shells of area 4πr2 and width dr to get volume.
 

1. What is the derivative of the area of a circle with respect to its circumference?

The derivative of the area of a circle with respect to its circumference is equal to half the circumference. This can be expressed mathematically as dA/dC = 1/2 * C, where A is the area and C is the circumference.

2. How is the derivative of area with respect to circumference used in real-world applications?

The derivative of area with respect to circumference is used in various real-world applications, such as calculating the rate of change of area with respect to circumference in circular motion or determining optimal dimensions for circular objects in engineering and design.

3. Can the derivative of area with respect to circumference be negative?

Yes, the derivative of area with respect to circumference can be negative. This indicates that the area is decreasing as the circumference increases. In other words, the circles are getting smaller.

4. How is the derivative of area with respect to circumference related to the derivative of circumference with respect to radius?

The derivative of area with respect to circumference is related to the derivative of circumference with respect to radius through the formula dA/dC = (dA/dr) * (dr/dC), where r is the radius. This formula shows that the derivative of area with respect to circumference is equal to the product of the derivative of area with respect to radius and the derivative of circumference with respect to radius.

5. Can the derivative of area with respect to circumference be used to find the area of a circle?

No, the derivative of area with respect to circumference cannot be used to find the area of a circle. The derivative only gives us information about the rate of change of area with respect to circumference, but it does not give us the actual value of the area. To find the area of a circle, we need to use the formula A = π * r^2, where r is the radius.

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