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In coordinate-free terms, the gradient of a function () may be defined by: d f = ∇ f ⋅ d r {\displaystyle df=\nabla f\cdot d\mathbf {r} } where d f {\displaystyle df} is the total infinitesimal change in f {\displaystyle f} for an infinitesimal displacement d r {\displaystyle d\mathbf {r} } , and is seen to be maximal when d r ...
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.
In color science, a color gradient (also known as a color ramp or a color progression) specifies a range of position-dependent colors, usually used to fill a region. In assigning colors to a set of values, a gradient is a continuous colormap, a type of color scheme. In computer graphics, the term swatch [1] has come to mean a palette of active ...
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term ...
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The gradient theorem states that if the vector field F is the gradient of some scalar-valued function (i.e., if F is conservative), then F is a path-independent vector field (i.e., the integral of F over some piecewise-differentiable curve is dependent only on end points). This theorem has a powerful converse:
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.