Cut vector, normal vector, gradient

Zookeeper

First, it indicates that the gradient is perpendicular to the cut vector of the curve, that is, the gradient direction is the direction of the normal vector:

Set the curve x = x (t), Y = Y (T), and z = z (T) to the surface u (x, y, z) = a curve on C (C is a constant, u (x, y, z) = C indicates the contour line). Because the curve is on the surface, x = x (t ), y = Y (t), Z = Z (t) satisfies the equation u (x, y, z) = C, that is, u (x (t), y (t ), Z (t) = C. Using the derivative law of the composite function, the two sides of the equation evaluate the derivative of T and get (EU/EX) * x' (t) + (EU/ey) * y' (t) + (EU/Ez) * Z' (t) = 0, so vector (x' (t ), y' (t), Z' (t) [tangent direction] is perpendicular to the vector (EU/ex, EU/Ey, EU/Ez) [gradient.

However, we need to note that the direction of the normal vector here is for the contour line, rather than for the curved surface. The method vector of the curved surface needs to be added to evaluate the U deviation. Because the gradient direction is dimensionality reduction, it is derived from the independent variables.

Vector (x' (t), y' (t), Z' (t) indicates the tangent vector of the curve, vector (EU/ex, EU/Ey, EU/Ez) indicates the gradient, so the gradient is perpendicular to the cut vector.

Speaking of this, it may not be easy to understand. We use mountain climbing as an example.

The fastest way to climb a mountain is to look at the steep direction. We can see the steep direction. When a mountain is climbing, the gradient direction is horizontal. You can't say the level when it extends to a high-dimensional dimension. That is, you can project the oblique direction to the horizontal plane, that is, the direction of the normal vector of the contour line.

In conclusion, when discussing the relationship between gradients, cut vectors, and normal vectors, the cut vectors are for curves rather than the whole surface's normal vectors. This is because the gradient dimension is more important than the curve dimension.

Cut vector, normal vector, gradient