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Dustinsfl
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How do I set up the following problem?
What geometric surface encloses the maximum volume with the minimum surface area?
What geometric surface encloses the maximum volume with the minimum surface area?
dwsmith said:How do I set up the following problem?
What geometric surface encloses the maximum volume with the minimum surface area?
ThePerfectHacker said:I think you want to rephrase it as, "What geometric surface encloses the maximum volume with a given surface area". In other words, you have 1 unit-squared of material and you want to enclose a surface to yields maximum volume.
Let $\Omega$ be bounded region in $\mathbb{R}^3$. You want to maximize,
$$ \iiint_\Omega 1 $$
Given the condition that,
$$ \iint_{\partial \Omega} 1 ~ ds = 1 $$
If we care about small details this question is a lot more complicated to phrase. Then we need to restrict ourselves to measurable sets such that .. blah blah blah.
HallsofIvy said:That's still a very badly phrased question. I think they expect you to combine the facts that the smallest surface containing a given volume is a sphere and that the largest area for a given size surface is a sphere.
The calculus of variation is a branch of mathematics that deals with finding the optimal value of a functional, which is a function that takes in other functions as inputs. It involves finding the function that minimizes or maximizes the value of the functional.
In the context of maximizing volume and minimizing area, the calculus of variation involves finding the function that maximizes or minimizes the volume or area, respectively, of a given shape or object. This is achieved by setting up an appropriate functional and using the Euler-Lagrange equation to find the optimal function.
The Euler-Lagrange equation is a necessary condition for finding the optimal solution to a variational problem. It states that the derivative of the functional with respect to the function being optimized must be equal to zero. This allows us to solve for the optimal function by solving the resulting differential equation.
The calculus of variation has many applications in various fields such as physics, engineering, economics, and biology. Some examples include finding the optimal path for a particle to travel from one point to another, minimizing the energy required for a system to reach equilibrium, and optimizing the shape of an airplane wing for maximum lift.
One limitation of the calculus of variation is that it can only be applied to problems that can be formulated as a functional. It also assumes that the optimal function exists and is smooth, which may not always be the case in real-world scenarios. Additionally, the solutions obtained through the calculus of variation may not always be unique, and there may be multiple optimal functions that satisfy the Euler-Lagrange equation.