limiting gradient

简明释义

极限坡度

英英释义

例句

1.In soil mechanics, understanding the limiting gradient is crucial for ensuring stability in slopes.

在土力学中，理解限制梯度对于确保坡度的稳定性至关重要。

2.The limiting gradient for this type of embankment is typically around 1:3.

这种类型的堤坝的限制梯度通常约为1:3。

3.During the construction project, we had to adjust our plans based on the limiting gradient of the terrain.

在施工项目中，我们不得不根据地形的限制梯度调整我们的计划。

4.The engineer calculated the limiting gradient to determine the maximum slope for the road design.

工程师计算了限制梯度以确定道路设计的最大坡度。

5.The limiting gradient helps in assessing the risk of landslides in hilly areas.

在丘陵地区，限制梯度有助于评估滑坡风险。

作文

In the realm of mathematics and physics, the concept of a limiting gradient is crucial for understanding how systems behave under certain conditions. The limiting gradient refers to the maximum rate of change of a function at a given point, especially as it approaches some boundary or limit. This idea can be applied in various fields, including calculus, optimization, and even in real-world scenarios such as engineering and economics.To illustrate the significance of the limiting gradient, consider a simple example involving a curve on a graph. As one moves along the curve, the slope or gradient changes. However, as one approaches a specific point, the gradient can approach a certain value, which we refer to as the limiting gradient. This value helps us understand the behavior of the curve near that point, providing insights into its characteristics, such as whether it is increasing or decreasing.In calculus, the concept of the limiting gradient is often expressed through derivatives. The derivative of a function at a point gives us the instantaneous rate of change, which is essentially the limiting gradient as we take smaller and smaller intervals around that point. For instance, if we have a function that models the height of a ball thrown into the air, the limiting gradient at the peak of its trajectory would indicate that the ball is momentarily at rest before it begins to fall back down. Understanding this point allows engineers to design better structures, ensuring they can withstand forces at critical points.Moreover, the limiting gradient has significant implications in optimization problems. When trying to find the maximum or minimum values of a function, we often set the derivative equal to zero to find critical points. These critical points are where the limiting gradient is zero, indicating a potential local maximum or minimum. By analyzing these points, we can determine the best solutions to various problems, from maximizing profit in a business to minimizing costs in production.In the context of real-world applications, the limiting gradient plays a vital role in fields such as economics and environmental science. For example, in economics, the limiting gradient could represent the marginal cost or marginal benefit of producing one more unit of a good. Understanding these gradients helps businesses make informed decisions about production levels and pricing strategies. Similarly, in environmental science, the limiting gradient can help model the rate of change in ecological systems, such as population growth or resource depletion, allowing for better management practices.In conclusion, the limiting gradient is an essential concept that transcends mathematical theory and finds practical applications across various disciplines. Whether in calculus, optimization, engineering, or economics, understanding the limiting gradient enables us to analyze and predict behaviors effectively. As we continue to explore complex systems and their interactions, the importance of grasping such concepts will only grow, highlighting the need for a solid foundation in both theoretical and applied mathematics.

在数学和物理的领域中，‘limiting gradient’（极限梯度）的概念对于理解系统在特定条件下的行为至关重要。‘limiting gradient’指的是在给定点上函数的最大变化率，尤其是在它接近某个边界或极限时。这个理念可以应用于多个领域，包括微积分、优化，甚至在现实世界场景中，如工程和经济学。为了说明‘limiting gradient’的重要性，考虑一个涉及图形上曲线的简单例子。当沿着曲线移动时，斜率或梯度会发生变化。然而，当接近某个特定点时，梯度可能会趋近于某个值，这就是我们所称的‘limiting gradient’。这个值帮助我们理解曲线在该点附近的行为，提供关于其特征的见解，例如它是增加还是减少。在微积分中，‘limiting gradient’的概念通常通过导数来表达。函数在某一点的导数给我们提供了瞬时变化率，本质上就是我们在该点周围取越来越小的区间时的‘limiting gradient’。例如，如果我们有一个模型描述抛向空中的球体高度的函数，那么在其轨迹的顶点处的‘limiting gradient’将表明球体暂时静止，然后开始下落。理解这一点使工程师能够设计出更好的结构，确保它们能够承受在关键点上的力量。此外，‘limiting gradient’在优化问题中具有重要意义。当试图找到函数的最大值或最小值时，我们通常将导数设为零，以找到临界点。这些临界点是‘limiting gradient’为零的地方，表明可能存在局部最大值或最小值。通过分析这些点，我们可以确定各种问题的最佳解决方案，从最大化企业利润到最小化生产成本。在现实世界应用的背景下，‘limiting gradient’在经济学和环境科学等领域发挥着重要作用。例如，在经济学中，‘limiting gradient’可以代表生产一单位商品的边际成本或边际收益。理解这些梯度帮助企业就生产水平和定价策略做出明智的决策。同样，在环境科学中，‘limiting gradient’可以帮助建模生态系统中的变化率，如种群增长或资源枯竭，从而实现更好的管理实践。总之，‘limiting gradient’是一个基本概念，超越了数学理论，并在多个学科中找到了实际应用。无论是在微积分、优化、工程还是经济学中，理解‘limiting gradient’使我们能够有效地分析和预测行为。随着我们继续探索复杂系统及其相互作用，掌握这样的概念的重要性只会增加，突显出在理论和应用数学中打下坚实基础的必要性。