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l = slope length α = angle of inclination. The grade (US) or gradient (UK) (also called stepth, slope, incline, mainfall, pitch or rise) of a physical feature, landform or constructed line is either the elevation angle of that surface to the horizontal or its tangent. It is a special case of the slope, where zero indicates horizontality. A ...
Slope illustrated for y = (3/2)x − 1.Click on to enlarge Slope of a line in coordinates system, from f(x) = −12x + 2 to f(x) = 12x + 2. The slope of a line in the plane containing the x and y axes is generally represented by the letter m, [5] and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line.
The gradient transforms like a vector under change of basis of the space of variables of . If the gradient of a function is non-zero at a point p {\displaystyle p} , the direction of the gradient is the direction in which the function increases most quickly from p {\displaystyle p} , and the magnitude of the gradient is the rate of increase in ...
Geothermal gradient is the rate of change in temperature with respect to increasing depth in Earth's interior. As a general rule, the crust temperature rises with depth due to the heat flow from the much hotter mantle ; away from tectonic plate boundaries , temperature rises in about 25–30 °C/km (72–87 °F/mi) of depth near the surface in ...
Stream gradient (or stream slope) is the grade (or slope) of a stream. It is measured by the ratio of drop in elevation and horizontal distance. [ 1 ] It is a dimensionless quantity , usually expressed in units of meters per kilometer (m/km) or feet per mile (ft/mi); it may also be expressed in percent (%).
A spatial gradient is a gradient whose components are spatial derivatives, i.e., rate of change of a given scalar physical quantity with respect to the position coordinates in physical space. Homogeneous regions have spatial gradient vector norm equal to zero.
If is a vector (a tensor of first degree), the gradient is a covariant derivative which results in a tensor of second degree, and the divergence of this is again a vector.
A linear function () = + has a constant rate of change equal to its slope a, so its derivative is the constant function ′ =. The fundamental idea of differential calculus is that any smooth function f ( x ) {\displaystyle f(x)} (not necessarily linear) can be closely approximated near a given point x = c {\displaystyle x=c} by a unique linear ...