Solving Raindrop Paths on Ellipsoid with 4x^2+y^2+4z^2=16

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Homework Help Overview

The problem involves an ellipsoid defined by the equation 4x^2 + y^2 + 4z^2 = 16, with a focus on determining the paths that raindrops would take as they slide down the surface under the influence of gravity. The discussion touches on concepts from vector calculus and physics.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the relevance of the gradient in determining the paths of raindrops, with some suggesting the use of the Euler-Lagrange equation to minimize time. Others express uncertainty about how to derive the curves from the gradient and question the necessity of specific starting points for the raindrops.

Discussion Status

The discussion is ongoing, with participants exploring different mathematical approaches and clarifying concepts related to gradients and paths of descent. Some guidance has been offered regarding the use of the gradient, but there is no explicit consensus on the method to be used.

Contextual Notes

Participants note that they have not yet covered certain mathematical techniques, such as the Euler-Lagrange equation, which may be relevant to the problem. There is also mention of needing to establish a specific point of origin for the raindrops on the ellipsoid.

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We have an ellipsoid with the equation 4x^2 + y^2+ 4z^2 = 16, and it is raining. Gravity will make the raindrops slide down the dome as rapidly as possible. I have to describe the curves whose paths the raindrops follow. This is probably more vector calculus than physics, but i wasn't sure how to solve it, so i don't know if knowledge of physics is necessary. I found the gradient, and i know that the rain will probably flow where the gradient is largest; however, i don't know how to go on.
 
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my guess is you need the Euler-Lagrange equation... trying to minimize the time, time=integral(ds/v)...
 
we haven't learned that quite yet, but thanks; I have a strong feeling the gradient of the surface is involved, but I don't see how one can get the curves by taking the path of maximum descent.
 
You can certainly answer this by finding the gradient...however, the drops will follow the path of the NEGATIVE gradient. To me, this should be a sufficient answer. To find a CURVE you would need to know the specific point from which you assume the rain drops originate on the ellipsoid (the top of the ellipsoid?) If the drops originate from the top of the ellipsoid just determine the direction of the gradient by plugging in the point. Then you can determine the line that describes the orientation of the gradient, which is a relationship between x and y. plug in y as a function of x into your original equation and solve for z. The result is z as a function of x, which represents a cross section of the ellipsoid.
 

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