The significance of gradient Grad = AI + BJ in gradient What kind of quantity is grad? Is it the rate of change on the z-axis? What is the corresponding relationship? Whenever X-axis changes a, Y-axis changes B, z-axis changes grad? Grad should be in the XY plane

The significance of gradient Grad = AI + BJ in gradient What kind of quantity is grad? Is it the rate of change on the z-axis? What is the corresponding relationship? Whenever X-axis changes a, Y-axis changes B, z-axis changes grad? Grad should be in the XY plane

If there is a binary function z = f (x, y), when it moves from point a to point B (let the moving distance be l), then the value of function Z has an increment M. when l tends to infinity, if M / L has a limit value, then this limit value is called the directional derivative of function in direction ab
After passing through point a, the function can move in any direction (of course, the moving range must be in the domain of definition), the function will have any number of directional derivatives, but the directional derivative in one direction must be the largest. This direction is represented by the vector of gradient (Grad = AI + BJ), where a is the partial derivative of the function in the X direction and B is the partial derivative of the function in the Y direction, The modulus of the gradient is the value of the maximum directional derivative

Please explain the geometric and physical meaning of gradient, curl and divergence

Let the physical parameter (such as temperature, velocity, concentration, etc.) in the system be w, and the parameter is w + DW at its vertical distance Dy, 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, The gradient of a scalar field is a vector field. The gradient at a certain point in the scalar field points to the fastest growing direction of the scalar field, and the length of the gradient is the maximum rate of change. More strictly speaking, the gradient of a function from RN to R in Euclidean space is the best linear approximation at a certain point in RN. In this sense, the gradient is a special case of Jacobian matrix, Gradient is only derivative, or, for a linear function, that is, the slope of a 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 product of the vector gradient and the direction under study. The value of the gradient is sometimes called gradient. In the case of binary function, let z = f (x, x), y) If there is a first order continuous partial derivative in the plane domain D, then for every point P (x, y) ∈ D, we can determine a vector (δ f / x) * I + (δ f / y) * J. this vector is called the gradient of function z = f (x, y) at point P (x, y), denoted as gradf (x, y). Similarly, we can define a three variable function: (δ f / x) * I + (δ f / y) * j + (δ f / z) * k denoted as grad [f (x, y, z)] gradient, which is originally a vector, When the directional derivative of a function along the direction at a certain point obtains the maximum value at that point, that is, the function changes fastest along the direction at that point, and the rate of change is the maximum (which is the modulus of the gradient). The mathematical definition of curl is that the projection of vector field a (x, y, z) = P (x, y, z) I + Q (x, y, z) j + R (x, y, z) K on the coordinate axis is δ R / δ Y - δ Q / δ Z, δ P / δ Z - δ R / δ x respectively, The vector of δ Q / δ X - δ P / δ y is called the curl of vector field a, which is recorded as rot a or curl a, that is, rot a = (δ R / δ Y - δ Q / δ z) I + (δ P / δ Z - δ R / δ x) j + (δ Q / δ X - δ P / δ y) k, where δ is a partial derivative sign, If a = ax · I + ay · j + AZ · K, then rota = (DZ / dy dy dy / DZ) I + (DAX / DZ DZ / DX) j + (Dai / DX DX / dy) k is a vector. The physical meaning of curl assumes that if the closed curve is reduced to a certain point within it, the area bounded by the closed curve l will gradually decrease, It has nothing to do with the shape of the closed curve, but obviously depends on the normal direction of the area bounded by the closed curve. Generally, the positive direction of L and the stipulation constitute the right-hand helix rule. The importance of curl lies in that it can be used to characterize the circulation strength of vectors in various directions near a certain point, When the volume Δ V defined by s approaches to zero in any way, the limit of the ratio ∮ f · DS / Δ V is called the divergence of vector field F at point m, and is recorded as div F, Div f represents the flux of vector f emitted from the unit volume at point m, so div f describes the density of flux source. The importance of divergence is that it can be used to characterize the divergence degree of vector field at each point in space. When div f > 0, it means that there is a positive source of flux emitted from the point; when div f > 0, it means that there is a positive source of flux emitted from the point

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

Divergence
Divergence refers to the change rate of unit volume of fluid in motion. In short, the concentrated area of fluid in motion is convergence, and the divergent area in motion is divergence. The quantity represented by divergence is called divergence. When the value is negative, convergence is beneficial to the development and enhancement of weather system. When the value is positive, divergence is beneficial to the dissipation of weather system. The physical quantity representing convergence and divergence is divergence
Curl, (formula can't be written here) see
gradient
gradient
If the physical parameters, such as the temperature gradient, the concentration gradient, the velocity, the distance and so on, are also called the physical parameters
In vector calculus, the gradient of a scalar field is a vector field. The gradient at 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 rate of change. More strictly speaking, the gradient of a function from the Euclidean space RN to R is the best linear approximation at a point in RN. In this sense, the gradient is a special case of the Jacobian matrix
In the case of a real valued function of a single variable, the gradient is only the derivative, or, for a linear function, the slope of the line
Gradient is sometimes used for slope, that is, the slope of a surface in a given direction. The slope can be obtained by taking the product of the vector gradient and the direction under study. The value of gradient is sometimes called gradient

What is the physical meaning of curl, divergence and annular flow gradient (in Engineering Electromagnetics), and how to prove the application
What is the physical meaning of divergence, curl, annular flux and gradient (in Engineering Electromagnetics) and how to prove it?

Divergence refers to the rate of change of unit volume when a fluid moves. If your field is a velocity field, the divergence of the field is the net flow of the fluid out of unit volume in unit time at a certain point. If the divergence of a field is not zero at a certain point, it means that the field is active at that point. For example, if the divergence of an electric field is not zero at a certain point, it means that there is an electric charge at that point, If divergence is negative, curl tells you that a field is integrating in a loop along an infinitesimal plane boundary, and the plane normal vector is given by curl vector, The length of the curl vector is the ring product per unit area. The circulation is related to the curl. The gradient is the spatial rate of change of the field. More specifically, you can read books on electrodynamics. For example, I remember the second volume of Feynman's lecture

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 means that there is a positive source of flux at that point

What is the negative gradient direction of a function?

The steepest descent direction of the potential function is its negative gradient direction

|a+b|

The shortest line between two points
Let the plane vector AB connect AB in order
The starting point P1 of a, the end point P2 of a, and the end point P3 of B
That is, vector p1p2 = a, p2p3 = B,
|A | and | B | are the lengths of vectors a and B,
Vector a + B = p1p3
Then | a + B | = | p1p3 |,
Then, according to the shortest line between two points
That is, p1p3|

|The geometric meaning of A-B |

Is the distance between the points represented by a and B on the number axis

Geometric meaning of λ (a-b) = λ a - λ B

The difference of two vectors is equal to the difference of two vectors

Geometric meaning of (a + b) (a-b) = a × A-B × B

The side length of a square is a
Cut off the edge of a small square, side length B
cut open
Put them together to get a rectangle of length a + B and width a-b
There's no way to draw