The original meaning of gradient is a vector (vector), which means that the directional derivative of a function at the point gets the maximum value along the direction, that is, the function at the point changes fastest along the direction (the direction of the gradient), with the maximum rate of change (the modulus of the gradient).
Let function of two variables If there is a first order continuous partial derivative on the plane region D, a vector can be determined for each point P (x, y) , this function is called a function The gradient at point P (x, y) is recorded as gradf (x, y) or , namely: among Is called a (two-dimensional) vector differential operator or Nabla operator ， 。 set up Is the unit vector in direction l, then Because when the direction l is consistent with the gradient direction
So when l is in the same direction as the gradient, Directional derivative Modulus with maximum value and gradient, namely Therefore, the function has the maximum rate of change along the gradient direction at a point, and the maximum value is the modulus of the gradient. [1]
The concept of gradient can be extended to the case of ternary functions.
Let the function of three variables It has the first order continuous partial derivative in the space region G, point , weighing vector Is a function The gradient at point P is recorded as or , i.e among Called a (three-dimensional) vector differential operator Or Nabla operator, 。 Similarly, the gradient direction is consistent with the direction of obtaining the maximum directional derivative, and its modulus is the maximum value of the directional derivative. [2]
Fig. 1 Expression of temperature gradient In vector calculus, Scalar field Of gradient Is a Vector field 。 Scalar field The gradient at a certain point in is pointing to the direction where the scalar field grows fastest, and the length of the gradient is the maximum rate of change. More strictly, from Euclidean space R n reach R The gradient of the function of is R n Best at a point Linear approximation 。 In this sense, the gradient is Jacobian matrix Special circumstances. In the case of single variable real valued functions, gradient just derivatives , or, for a linear function , that is, linear Slope 。 gradient The word is sometimes used Slope , that is, a surface along a given direction tilt Degree. By taking the vector gradient and Dot product Get the slope. The value of gradient is sometimes called gradient. 
