table of Contents:
definition
One Function gradient
Binary function gradient
Ary function gradient
Gradient derivative
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First, first look at the most common definition:
Gradient is a vector, he always points to the direction of the largest fastest rate of change.
Gradient direction always points to a greater value function w direction.
Concept can be applied to the gradient of mono-, di-, tri- or more functions of the function argument, but the most common is the first three.
In a univariate function of:
In the real-valued function of one variable, the gradient just derivative , or for a linear function , which is the line slope .
In the binary function:
In binary function, the gradient is based on the concept of a gradient of the partial derivatives of the direction based on the concept of derivative. Directional derivative do not look, feel intuitively first binary function of the gradient.
Binary function: W (x, y) = x ^ 2 + y ^ 2;
Gradient: <2x, 2y>, into the required points (x and y are partial derivatives: ∂z / ∂x = 2x derivative of the function represented by the X-axis direction while ∂z / ∂y = 2y represents a function derivative of the Y-axis direction.)
Function of the image as shown in FIG. 2.1, then as a gradient vector of the pointing direction of what it?
So that W (x, y) equal to a constant C, to obtain a projected view is a projection of concentric circles. <2x, 2y> we can get on with projectors according to two important conclusions:
1> binary function gradient direction parallel to the xy plane is formed.
2> direction perpendicular to the concentric gradient, i.e. perpendicular to the contour.
3> gradient perpendicular to the tangential direction (skip proof)
FIG. 2.1 f (x, y) = x ^ 2 + y ^ 2-dimensional FIG.
FIG. 2.2 f (x, y) = x ^ 2 + y ^ 2 projection
Three-way function:
Order in the ternary function W (x, y, z) = c, a curved surface would be obtained.
Ary function: W (x, y, z) = x ^ 2 + y ^ 2 + y ^ 2;
Gradient: <2x, 2y, 2z>
1> gradient as a vector in three-dimensional space.
2> direction perpendicular to the slice gradient direction, also the direction of the gradient vector and the curved surface section method, can be used to solve the plane equation.
Directional derivative
We also mentioned above, the directional derivative, a so-called directional derivative is a function of the domain within the point, the demand for the pilot to obtain a certain direction derivative . Typically binary function directional derivative of the function and ternary direction derivative, directional derivative can be divided in the linear direction and the direction of the curve, and here we talk binary function.
Let's look at the conclusions:
When the same directional derivative and gradient direction, the original function of the fastest growing.
Direction opposite to the gradient direction and the time derivative of the original function to reduce the fastest.
When the gradient and directional derivative in the direction orthogonal to the original function is not to rise.
Let's look at the directional derivative (observe Figure 4-1), introduced by the above binary function we know along ∇W (gradient direction) the direction of maximum rate of change of the fastest growth function, that function of the rate of change along the other direction how should we change how should we say then?
Pic 4-1
Suppose we now have the function W (x, y), we know that along the x, y directional derivative how demand. Now suppose there is now a unit vector u, u direction along the derivative of what is it? (U = cosθi + sinθj, a unit vector)
Figure 4-2
Unfinished to be updated
