n. 1a. the slope of a line measured as the ratio of its vertical change to its horizontal change. Thus the gradient of the line joining points (x₁, y₁) and (x₂, y₂) in the Cartesian plane, as shown in figure (a) is and the gradient form of the equation of a line through the point (x₁, y₁) is that is, where the equation of a line is given as y = mx + c, m is the gradient of the line, and c its intercept with the y-axis.
b. more generally, the slope of a curve at a point measured as the slope of the tangent at that point; the gradient of the curve for x = a is the instantaneous rate of change in the value of the function. This is given by the limit, as x approaches a, of the ratio of the change, Δy, in the dependent variable to the change, Δ x, in the independent variable, as shown in figure (b) above. derivative. 2. the vector whose components parallel to each coordinate axis are the partial derivatives of a given function with respect to the variable mapped on that axis, and whose direction is that in which the derivative of the function has its maximum value; the vector Often it is necessary that the partials be continuous, in which case the gradient is identifiable with the derivative of the vector function. The gradient is written gradF or ∇F. See also Frechet derivative, curl, divergence. 3. more generally, for a Cartesian tensor
T_ijk... e_i ⊗ e_j ⊗ e_k ⊗ ...
the quantity
∂/∂ x_p (T_ijk...) e_i ⊗ e_j ⊗ e_k ⊗ ... ⊗ e_p.