**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Slope

Summary

In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m; there is no clear answer to the question why the letter m is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as and it can also be found in Todhunter (1888) who wrote it as "y = mx + c".
Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical – as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan.
The steepness, incline, or grade of a line is measured by the absolute value of the slope. A slope with a great

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related publications (100)

Loading

Loading

Loading

Related people (19)

Related concepts (47)

Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate

Derivative

In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the p

Calculus

Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
It has two major br

Related courses (50)

MGT-581: Introduction to econometrics

The course provides an introduction to econometrics. The objective is to learn how to make valid (i.e., causal) inference from economic data. It explains the main estimators and present methods to deal with endogeneity issues.

MGT-454: Principles of microeconomics

The course allows students to get familiarized with the basic tools and concepts of modern microeconomic analysis. Based on graphical reasoning and analytical calculus, it constantly links to real economic issues.

MATH-106(a): Analysis II

Étudier les concepts fondamentaux d'analyse, et le calcul différentiel et intégral des fonctions réelles de plusieurs
variables.

Related units (5)

Related lectures (117)

Sediment transport is one of the main morphodynamic processes in mountain streams. Understanding bedload transport is fundamental to prevent and mitigate hydrogeological hazards and for the resources management. The classic bedload discharge predictions are usually expressed by empirical formulae as a one-to-one relationship between sediment flux and water discharge. Unfortunately, this framework is often not sufficient to describe the variability of bedload discharge. In fact, large fluctuations in bedload transport rates are reported, even for steady flow conditions, over different time scales. Especially in case of mountain streams, the predicted and measured bedload transport rates may reflect differences of several orders of magnitude. Different processes, acting at different spatial and temporal scales, have been proposed as causes of the observed fluctuations, such as the intrinsic variability of particle paths, the migration of bedforms or the unsteadiness in sediment supply. During the second half of the twentieth century, some scientists started to study the bedload transport as a stochastic process involving the grains transported by a stream. The purpose of my research is to assess and improve the performance of a stochastic model in conditions representative of mountain streams and in presence of bedforms. To address this challenge, experiments will be performed in a narrow 2D flume, with steady flow conditions and stationary sediment feeding, for slope angles varying between 1 and 3°. The sediment feeding rate and the bedload tranport rate at the outlet will be measured using high-speed cameras and image processing techniques. Another high-speed camera will acquire side images of the flume for monitoring the evolution of the bed topography and for tracking the moving particles inside a control window. The analyses will be focused on activity and velocity distribution of the particles inside the control window. The collected data will allow the comparison of model outputs with the observed fluctuations of bedload discharge. The dependence of this fluctuations on different variables (shear stress, slope, antidune geometry, antidune migration celerity, etc.) will be investigated.

2018Application of stepped spillways increases the energy dissipation rate along the spillway and may reduce the dimensions of the terminal energy dissipation structure. This pronounced energy dissipation makes stepped chutes attractive under various conditions, namely as service spillways on RCC gravity dams and on valley flanks near earth dams. For both, in some cases, an abrupt slope change may be required to be implemented on stepped chutes in order to follow the site topography and to minimize the needed excavations and hence respective costs. An abrupt slope change along stepped spillways can influence the flow properties such as the air entrainment, velocity and pressure distribution, and the energy dissipation. A quite limited number of stepped spillways have been built with an abrupt slope change, whereas no systematic scientific investigation for designing such type of configuration has been conducted to date. Accordingly, comprehensive information on the effect of an abrupt slope change on the flow features is missing. Therefore, the present experimental research work aimed to examine the effect of an abrupt slope change (from steep to mild) on the skimming flow features, by analysing the air entrainment, flow bulking, velocity and dynamic pressure development and energy dissipation along the stepped chute. Physical modelling was conducted in a relatively large scale facility with slope changes from 50º to 18.6º and 50º to 30º. Detailed air-water flow measurements were conducted at several cross-sections (step edges) along the chute, upstream and downstream of the slope change. In addition, dynamic pressure measurements were obtained on both vertical and horizontal faces of several steps in the vicinity and far downstream of slope change cross-section. The results indicated a substantial influence of abrupt slope changes on the flow properties for the tested range of flow rates, particularly in comparison with typical results for constant sloping stepped spillway flows. Four main local sub-regions have been found to describe the typical air-water flow patterns in the vicinity and further downstream of the slope change, namely with regard to the mean (depth-averaged) air concentration, air concentration distribution, pseudo-bottom air concentration, air-phase frequency and characteristic flow depths. The relative head loss corresponding to the reach under the influence of the slope change was found to vary between 38% to 51%, for the tested range of flow rates and slope change configurations. Mean pressures up to approximately 21 times the equivalent clear water depth (approximately 13 times the step height) were observed on the horizontal step faces in the vicinity of slope change cross-section for the tested range of flow rates. In conclusion, for the first time, the influence of an abrupt slope change on skimming flow properties on stepped spillways was investigated with systematic experiments on two slope change configurations and for a wide range of relative critical flow depths. This thesis report describes and discusses the achieved results mainly on the air entrainment and flow bulking, velocity and dynamic pressure distributions, as well as the energy dissipation.

Stepped spillways have been built since several decades in combination with roller compacted concrete dams. More recently stepped spillways are also excavated into rock along the abutments of embankment dams. According to the prevailing topography these stepped spillways are designed with variable step heights and slope changes along the channel. In his research project Dr Mohammad Javad Ostad Mirza studied for the first time systematically with laboratory tests the influence of abrupt slope changes on the flow characteristics over stepped spillways. The air-water flow behaviour was studied in detail at several cross-sections along the chute, upstream and downstream of the slope change by measuring the evolution of water surface based on the equivalent clear water depth, flow bulking, flow velocities and air concentration profiles. Furthermore, dynamic pressures were measured on both vertical and horizontal faces at several steps in the vicinity and far downstream of the slope change. The systematic experiments give new insights in the flow characteristics over stepped spillways in the vicinity of slope changes, which is helpful for practical applications.