Derivative and gradient, tangent and Normal Vector

From: http://www.cnblogs.com/jerrylead/archive/2011/03/09/1978280.html

The author makes it very clear

Remember that the tangent of the curve is often required when you are doing a math problem in high school. If you see a function like this, the slope of the tangent is obtained regardless of the result obtained by the derivation of November 21, and the tangent at the position is obtained.

After going to college, I learned how to find the tangent of the curved surface and the method of the normal vector, and then used the formula to find the tangent.

A classic example is as follows:

(From a geometric application PPT on the Web)

Where the vector n is the partial derivative of f (x, y, z.

However, the two methods seem unrelated. The tangent is obtained in the middle. However, after the partial derivation, the method vector is obtained. Why is the difference so big? Why is the slice equation related to the normal vector?

Of course, questions and answers to these questions can be obtained through strict mathematical derivation. Here we want to explain the truth from a more straightforward perspective.

first, the normal vector (gradient) is f (x) (where x = {x0, x1, x2 ,... Xn} is the result of partial derivation of each component in an n-dimensional vector, representing the change rate of f (x) in each direction. The whole normal vector is f (x) vector superimposed by the rate of change in each direction. For example, for a one-dimensional f (x) =, the derivative of X is 2X, which means that the X direction changes at 2x speed. For example, when x = 2, F (x) the change rate is 4 greater than the change rate when x = 1 (the change rate is 2). The direction of the normal vector can only be X, because f (x) is one-dimensional. F (x) is called a hidden function. For example, if we use a hidden function, it can be expressed as f (x, y) = f (x)-y. In this way, f (x, y) is two-dimensional. As to why the derivative is the rate of change, we can know through the definition of the derivative (How much dy changes are caused by small DX changes ).

We understand that the normal vector of the implicit function f (x) is the partial derivative vector of f (x) to each component. So why are we looking for a tangent instead of a normal vector? In fact, we cannot confuse the Implicit Functions f (x) and. A implicit function is a function whose values vary according to the values of X. But only the constraint relationships between x and y are met. For example, if the x-y coordinate is established, the constraint relationships between the two can be expressed by graphs (straight lines, curves, etc. For example, we can use it to represent a parabolic curve and draw it in the x-y coordinate system. In other words, the implicit function is f (x, y) =. Only when f (x, y) is equal to a given value (for example, 0) is a parabolic, otherwise, it is just a function. If Z is used to replace f (x, y), F (x, y) is actually a surface, and the dimension increases by 1. The result of our partial derivation for F (x, y) is actually the change rate of the value Z of f (x, y.

Explain how much the value of f (x, y) will change in a small range of (x, y, this change rate is determined by the linear combination of the tiny transformation dx in the X direction and the tiny transformation dy in the Y direction, and their coefficients are partial derivatives. Replace Dx and Dy with the unit vectors I and j, which are the normal vectors. Then the gradient reflects the change rate and transformation direction of f (x) at a certain point.

In short, for an implicit function f (x), we want to know the variation direction and size of f (x) near a given X. How to portray it? Because X is in all directions (x0, x1, x2... XN) the change rate and direction are different (for example, x0 is changed by square level and X1 is changed by linear mode. This depends on the specific expression ), however, we want to know how they change together. We use the fully differential formula (for example, we can know that the superposition coefficient between them is the partial derivative, and the superposition result is the change rate, and the direction is x0, x1, x2... Corresponding change direction I, J, K... And other linear combinations to obtain the direction.

Return to the question of why the question "What is obtained in the center is a tangent". In fact, this is the final conclusion and is derived. In the first step, we will write the implicit function (here, X and Y are all real numbers, and the above X is a vector ),.

Then evaluate F's partial direction to X =

Evaluate the F-to-y deviation-1.

That is, the gradient is

Because the tangent and the normal vector are vertical, the inner product of the tangent and the normal vector is 0.

Set the tangent direction vector to (m, n.

It can be seen that the tangent slope is.

Return to the surface in the blue image above to find the tangent plane. After finding the normal vector of a certain point, the tangent plane of the point must meet two conditions. First, the tangent point must be crossed, the change direction of the point is to be reflected (this is not the change direction of the f (x) value of the point, but the change direction of the point itself ). However, the variation of the point eventually reflects the variation of the f (x) value of the point, that is, the variation of the cut plane reflects the variation of the normal vector, and the partial derivative reflects the f (x) value. Therefore, the partial derivative of the tangent plane is the same as that of f (x. We can see from the blue image that the tangent plane uses the partial derivative of f (x.

In the above fully differential formula, we can better understand the extreme value. Why do we often say that the derivative of a function is 0 when it obtains the extreme value. Suppose it is a one-dimensional situation. The minimum value is required. After the two sides are divided, when X is 0, the derivative 2x is 0 to obtain the extreme value. Otherwise, if X is a positive number, DX only needs to be adjusted to the left (DX <0) to reduce the value of f (x). If X is a negative number, so DX only needs to adjust to the right (dx> 0) to make f (x) smaller. Therefore, the final adjustment result is x = 0. For two-dimensional situations,

The value of is positive and negative after calculation, but we should note that DX can be positive and negative, Dy can also be positive and negative, as long as one is not 0, then by adjusting dx, dy's plus and minus signs (that is, determine how to move X and Y) can make the value larger and smaller. Only when the partial derivative is 0, no matter how you adjust Dx and Dy, It is 0 to obtain the extreme value.

The above is just a simple understanding of some people. The purpose is to establish perceptual knowledge, which may cause some flaws.

* The following is a more general understanding of individuals:

If the function is a one-dimensional variable, the gradient is the direction of the tangent.

if it is greater than one dimension, the gradient is the normal vector at this point and points to the contour with a higher value, this is why the negative gradient is used when the minimum value is obtained.