Chinese name: gradientmbth: gradientdiscipline: calculus scope of application: related concepts of mathematical science: directional derivative properties: vector definition, popularization, application and definition If a binary function has a first-order continuous partial derivative in a plane region D, a vector can be determined for each point P(x, y). This function is called the gradient of the function at point P(x, y), and is denoted as gradf(x, y) or, that is, gradf(x, y)= = where it is called (two-dimensional) vector differential operand or Nabla operand. Let it be the unit vector in the L direction, then when the L direction is consistent with the gradient direction, the directional derivative has a maximum value, which is the modulus of the gradient, that is to say, the rate of change of the function along the gradient direction at a point is the maximum, and this maximum value is the modulus of the gradient. The concept of generalized gradient can be extended to the case of ternary function. Let a ternary function have a first-order continuous partial derivative at point G and point A in the space region, and call the vector the gradient of the function at point P, and write it as or, that is, = = It is called (three-dimensional) vector differential operator or Nabla operator here. Similarly, the direction of the gradient is consistent with the direction of obtaining the maximum directional derivative, and its modulus is the maximum value of the directional derivative. If the physical parameters (such as temperature, speed, concentration, etc. ) somewhere in the system is w, and at the place perpendicular to its dy is w+dw, which is called the gradient of physical parameters, that is, the rate of change of physical parameters. If the parameters are velocity, concentration, temperature or space, they are called velocity gradient, concentration gradient, temperature gradient or space gradient respectively. The expression of temperature gradient in cartesian coordinate system is shown on the right. Expression of temperature gradient In vector calculus, the gradient of scalar field is a vector field. The gradient of a point in the scalar field points to the fastest growing direction of the scalar field, and the length of the gradient is the maximum change rate. More strictly speaking, the gradient of a function from Euclidean space R n to R is the best linear approximation of a point in R n. In this sense, the gradient is a special case of Jacobian matrix. In the case of unary real function, the gradient is only a derivative, or for linear functions, it is the slope of a straight line. The term gradient is sometimes used to indicate slope, that is, the degree of inclination of a surface in a given direction. The slope can be obtained by the point product of the direction gradient and the research direction. The value of a gradient is sometimes called a gradient.