Minimizing Curve: Solving the Problem

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In summary: Using the chain rule, the derivatives of $u_1$ and $u_2$ can be written as \begin{align}\frac{du_1}{ds} &= \frac{\partial u_1}{\partial s} + \frac{\partial u_1}{\partial s}\frac{\partial s}{\partial u_1} \\ \frac{du_2}{ds} &= \frac{\partial u_2}{\partial s} + \frac{\partial u_2}{\partial s}\frac{\partial s}{\partial u_2}\end{align}Substituting these into
  • #1
Kreizhn
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I've been given a continuously differentiable function C(u1, u2) in 2-D Euclidean space. Now i have that for u a function of s, that
[tex] \frac{ dC}{ds} = \frac{\partial C}{\partial u_1}\frac{ du_1}{ds} + \frac{ \partial C}{\partial u_2} \frac{ du_2}{ds} [/tex]
which is obviously just an extension of the chain rule. Now I'm told to minimize this subject to the constraint that
[tex] \left( \frac{\partial u_1}{\partial s} \right)^2 + \left( \frac{\partial u_2}{\partial s} \right)^2 = 1 [/tex]
and given that the answer is
[tex] \frac{ du}{ds} = - \frac1{\left\| \frac{\partial C}{\partial u} \right\| } \frac{\partial C}{\partial u} [/tex]

My problem is that I don't see how we arrived at the answer. What am I missing?
 
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The answer is derived from a technique called Lagrange Multipliers. The idea is to use a constraint on the system to form a new function which can be minimized. In this case, the constraint is the given equation: $\left( \frac{\partial u_1}{\partial s} \right)^2 + \left( \frac{\partial u_2}{\partial s} \right)^2 = 1$. This can be rewritten as a function of $L(u_1,u_2,\lambda)$ such that \begin{align}L(u_1,u_2,\lambda) &= C(u_1,u_2) + \lambda \left( \frac{\partial u_1}{\partial s} \right)^2 + \lambda \left( \frac{\partial u_2}{\partial s} \right)^2 - \lambda \\ &= C(u_1,u_2) + \lambda \left[ \left( \frac{\partial u_1}{\partial s} \right)^2 + \left( \frac{\partial u_2}{\partial s} \right)^2 - 1 \right]\end{align}The value of $\lambda$ is chosen such that it enforces the constraint by making the expression in brackets equal to zero. Thus, the minimum of $L$ is equivalent to the minimum of $C$, subject to the constraint. Taking the partial derivatives of $L$ with respect to $u_1$ and $u_2$ yields \begin{align}\frac{\partial L}{\partial u_1} &= \frac{\partial C}{\partial u_1} + 2\lambda \frac{\partial u_1}{\partial s}\frac{\partial^2 u_1}{\partial s^2} \\\frac{\partial L}{\partial u_2} &= \frac{\partial C}{\partial u_2} + 2\lambda \frac{\partial u_2}{\partial s}\frac{\partial^2 u_2}{\partial s^2} \end{align}The goal is to find $\frac{du_1}{
 

1. What is meant by "minimizing curve" in scientific research?

In scientific research, minimizing curve refers to the process of reducing or decreasing the variability in data points in order to identify a clear trend or pattern. This is done through various statistical methods and techniques.

2. Why is it important to solve the problem of minimizing curve?

Solving the problem of minimizing curve is important because it allows scientists to accurately interpret and analyze their data, leading to more reliable and valid results. It also helps to identify any potential confounding factors or errors in the data.

3. What are some common methods for minimizing curve in research?

Some common methods for minimizing curve in research include using regression analysis, controlling for extraneous variables, and increasing the sample size. Additionally, using more precise and accurate measurement tools and techniques can also help to minimize curve.

4. How does minimizing curve impact the validity of scientific findings?

Minimizing curve is crucial for ensuring the validity of scientific findings. If there is a lot of variability in the data, it becomes difficult to draw accurate conclusions and make reliable predictions. By minimizing curve, scientists can increase the confidence in their findings and make more accurate interpretations.

5. Are there any drawbacks to minimizing curve in scientific research?

While minimizing curve is generally beneficial in scientific research, there can be some drawbacks. For example, if the researcher is too focused on minimizing curve, they may overlook important outliers or unique data points that could provide valuable insights. Additionally, some methods of minimizing curve may alter the original data, making it less representative of the true population.

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