Higher number derivative problem (derivative) stem "f (x) = x ^ 3-3ax ^ 2 + 2bx

Higher number derivative problem (derivative) stem "f (x) = x ^ 3-3ax ^ 2 + 2bx

X ^ 3 is the third power of X
f'(x)=x^3-3ax^2+2bx
=3x^2-6ax+2b

Divergence, curl, gradient What is their definition? Please write it down,

divergence
Divergence refers to the change rate of unit volume when a fluid moves. In short, the area where the fluid is concentrated in motion is convergence, and the area where the fluid is divergent in motion is divergence. The quantity expressed is called divergence. When the value is negative, it is convergence, which is conducive to the development and enhancement of the weather system. When it is positive, it is divergence, which is conducive to the dissipation of the weather system. The physical quantity indicating convergence and divergence is divergence
Curl, (the formula cannot be written here) see
gradient
gradient
Let the physical parameter (such as temperature, velocity, concentration, etc.) somewhere in the system be w, and the parameter is w + DW at dy perpendicular to it, then it is called the gradient of the physical parameter, that is, the change rate of the physical parameter. If the parameter is velocity, concentration or temperature, then it is called velocity gradient, concentration gradient or temperature gradient respectively
In vector calculus, the gradient of a scalar field is a vector field. The gradient at a certain point in the scalar field points to the direction of the fastest growth of the scalar field, and the length of the gradient is the maximum rate of change. More strictly, the gradient of the function from the Euclidean space RN to R is the best linear approximation at a certain point of RN. In this sense, the gradient is a special case of the Jacobian matrix
In the case of a univariate real valued function, the gradient is just a derivative, or, for a linear function, the slope of the line
The term gradient is sometimes used for slope, that is, the slope of a surface along a given direction. The slope can be obtained by taking the dot product of the vector gradient and the direction under study. The value of the gradient is sometimes called gradient

Geometric or physical meaning of divergence and curl in Calculus

Divergence: it can be used to characterize the divergence degree of vector field at each point in space. When div f > 0, it indicates that there is a positive source of radiative flux at this point; When div f

Geometric meaning of multivariate function gradient?

If the multivariate function is regarded as the height, its gradient is the steepest uphill direction
If the multivariate function is regarded as potential energy, the negative value of its gradient is the local force on the object

What is the geometric meaning of the "value" of the module of the gradient? Of course, he represents the maximum value of the directional derivative of the function. Does personal feeling (for example) represent the "slope" of the hillside in the steepest direction at a certain point, are you sure?

Your understanding is correct:
"Slope of a hillside in the steepest direction at a point"

Please explain the geometric meaning of gradient

Taking the binary function f (x, y) as an example, firstly, the gradient of F (x, y) at a point (x0, Y0) is a vector, and its direction is the direction in which the function value of function f (x, y) changes fastest at that point, that is, the direction in which the directional derivative is the largest, and its modulus is equal to the maximum value of the directional derivative at that point